∧I Φ`φ Φ`ψ Φ`φ∧ψ

∧I

∧ T F F F

φ T T F F

ψ T F T F

∧Er Φ`φ∧ψ Φ`φ

∧Er

φ T T F F

∧ T F F F

ψ T F T F

∧El

φ T T F F

∧ T F F F

ψ T F T F

∧El Φ`φ∧ψ Φ`ψ >I Φ`> Φ`> ⊥E Φ`⊥ ⊥

∧I

⊥E

> T ⊥ F

∨Il φ Φ`φ φ∨ψ Φ`φ∨ψ

∨Il

φ T T F F

∨ T T T F

ψ T F T F

φ T F T F ∨E

∨ T T T F

ψ T T F F

∨Ir ψ Φ`ψ φ∨ψ Φ`φ∨ψ

∨Ir

φ∨ψ Φ ` φ ∨ ψ, φ ` X , ψ ` X Φ`X

φ .. . X X

1

1

ψ .. . X

1

1 ∨E

∨E

→I 1

φ .. . ψ Φ`φ`ψ φ→ψ Φ`φ→ψ →E

Φ ` φ → ψ, φ Φ`ψ

φ→ψ ψ

φ T F F T

ψ T T F F

φ

φ→ψ T T T F φ T F T F

ψ T F F T

φ`ψ ` ` ` 0 φ→ψ T T F F

φ`ψ T T F F

φ T T F T F F F F

∨ T T T T T F F F

ψ T F T F T F F F

φ`X ` ` ` 0 ` ` ` `

ψ`X ` ` ` ` 0 ` ` `

X T T T F F T F F

¬I

Φ`φ`⊥ Φ ` ¬φ ¬E

φ .. . ⊥ ¬φ

φ F T F T

φ F F T T

¬φ

φ Φ ` ¬φ, φ Φ`⊥

¬φ T F F T



φ`⊥ T 0 0 0 ¬φ T F F T

Φ`⊥ ` ` 0 0

⇐⇒ I φ ψ .. .. . . ψ φ Φ ` φ ` ψ, ψ ` φ φ ⇐⇒ ψ Φ ` φ ⇐⇒ ψ ⇐⇒ El

Φ ` φ ⇐⇒ ψ, φ Φ`ψ

φ ⇐⇒ ψ ψ

φ T F F T

φ`ψ ` ` 0 `

ψ`φ ` ` ` 0

φ T T F F

ψ T F F T

φ ⇐⇒ ψ T F T F

φ T T F F

ψ T F F T

φ ⇐⇒ ψ T F T F

ψ T F T F

φ

⇐⇒ Er

Φ ` φ ⇐⇒ ψ, ψ Φ`φ

φ ⇐⇒ ψ φ

ψ

RAA

Φ ` ¬φ ` ⊥ Φ`φ

¬φ .. . ⊥ φ

φ T F F

¬φ F F T

¬φ ` ⊥ ` ` 0

φ ⇐⇒ ψ ` ` 0 0

Page 1 Ho ΦΕφ ΦΕψ 0 +012 ^ O A U т т т T F F F FT F F A Er ΦΕφΛύ ...

TTK. E. 70. 0 +0FI. 0570. E ф. -ф. E-0,0. QE1. 0 0 0 F1. Е Т Е. F F F. Т Е Е. TTK. AI. 9. O E ... o toe... FECT. TRII. Fly k. ¢ ¢ ¢ A. : : T T F F. F F E E. ° F T K. ΦΕφEψ ...

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