Theorem pfbycont-P1.16 86

Description: Proof by Contradiction.

Assertion

Ref	Expression
pfbycont-P1.16	⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))

Proof of Theorem pfbycont-P1.16

Step	Hyp	Ref	Expression
1		ax-L1 11	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → (𝜑 → 𝜓)))
2		trnsp-P1.15a 76	. . . 4 ⊢ ((𝜑 → ¬ 𝜓) → (𝜓 → ¬ 𝜑))
3	2	axL1.SH 30	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → (𝜓 → ¬ 𝜑)))
4	1, 3	sylt-P1.9.2AC.2SH 63	. 2 ⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → (𝜑 → ¬ 𝜑)))
5	4	nclav-P1.14.2AC.SH 75	1 ⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 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:  pfbycont-P1.16.AC.2SH  87