Theorem pfbycase-P1.17 88

Description: Proof by Cases.

Assertion

Ref	Expression
pfbycase-P1.17	⊢ ((𝜑 → 𝜓) → ((¬ 𝜑 → 𝜓) → 𝜓))

Proof of Theorem pfbycase-P1.17

Step	Hyp	Ref	Expression
1		trnsp-P1.15b 78	. . . 4 ⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))
2	1	axL1.SH 30	. . 3 ⊢ ((𝜑 → 𝜓) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
3		ax-L1 11	. . 3 ⊢ ((𝜑 → 𝜓) → ((¬ 𝜑 → 𝜓) → (𝜑 → 𝜓)))
4	2, 3	sylt-P1.9.2AC.2SH 63	. 2 ⊢ ((𝜑 → 𝜓) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜓)))
5	4	clav-P1.12.2AC.SH 70	1 ⊢ ((𝜑 → 𝜓) → ((¬ 𝜑 → 𝜓) → 𝜓))

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:  pfbycase-P1.17.2SH  89  pfbycase-P1.17.AC.2SH  90