Theorem axL3.AC.SH 47

Description: Deductive Form of ax-L3 13.

Hypothesis

Ref	Expression
axL3.AC.SH.1	⊢ (𝛾 → (¬ 𝜑 → ¬ 𝜓))

Assertion

Ref	Expression
axL3.AC.SH	⊢ (𝛾 → (𝜓 → 𝜑))

Proof of Theorem axL3.AC.SH

Step	Hyp	Ref	Expression
1		axL3.AC.SH.1	. 2 ⊢ (𝛾 → (¬ 𝜑 → ¬ 𝜓))
2		ax-L3 13	. . . 4 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))
3	2	axL1.SH 30	. . 3 ⊢ (𝛾 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
4	3	rcp-FR1.SH 40	. 2 ⊢ ((𝛾 → (¬ 𝜑 → ¬ 𝜓)) → (𝛾 → (𝜓 → 𝜑)))
5	1, 4	ax-MP 14	1 ⊢ (𝛾 → (𝜓 → 𝜑))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem is referenced by:  clav-P1.12  68