Theorem pfbycont-P1.16.AC.2SH 87

Description: Deductive Form of pfbycont-P1.16 86.

Hypotheses

Ref	Expression
pfbycont-P1.16.AC.2SH.1	⊢ (𝛾 → (𝜑 → 𝜓))
pfbycont-P1.16.AC.2SH.2	⊢ (𝛾 → (𝜑 → ¬ 𝜓))

Assertion

Ref	Expression
pfbycont-P1.16.AC.2SH	⊢ (𝛾 → ¬ 𝜑)

Proof of Theorem pfbycont-P1.16.AC.2SH

Step	Hyp	Ref	Expression
1		pfbycont-P1.16.AC.2SH.2	. 2 ⊢ (𝛾 → (𝜑 → ¬ 𝜓))
2		pfbycont-P1.16.AC.2SH.1	. . . 4 ⊢ (𝛾 → (𝜑 → 𝜓))
3		pfbycont-P1.16 86	. . . . . 6 ⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))
4	3	axL1.SH 30	. . . . 5 ⊢ (𝛾 → ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑)))
5	4	rcp-FR1.SH 40	. . . 4 ⊢ ((𝛾 → (𝜑 → 𝜓)) → (𝛾 → ((𝜑 → ¬ 𝜓) → ¬ 𝜑)))
6	2, 5	ax-MP 14	. . 3 ⊢ (𝛾 → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))
7	6	rcp-FR1.SH 40	. 2 ⊢ ((𝛾 → (𝜑 → ¬ 𝜓)) → (𝛾 → ¬ 𝜑))
8	1, 7	ax-MP 14	1 ⊢ (𝛾 → ¬ 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:  ndnegi-P3.3  168