Theorem bicmb-P2.5c.AC.2SH 121

Description: Deductive Form of bicmb-P2.5c 119.

Hypotheses

Ref	Expression
bicmb-P2.5c.AC.2SH.1	⊢ (𝛾 → (𝜑 → 𝜓))
bicmb-P2.5c.AC.2SH.2	⊢ (𝛾 → (𝜓 → 𝜑))

Assertion

Ref	Expression
bicmb-P2.5c.AC.2SH	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Proof of Theorem bicmb-P2.5c.AC.2SH

Step	Hyp	Ref	Expression
1		bicmb-P2.5c.AC.2SH.2	. 2 ⊢ (𝛾 → (𝜓 → 𝜑))
2		bicmb-P2.5c.AC.2SH.1	. . . 4 ⊢ (𝛾 → (𝜑 → 𝜓))
3		bicmb-P2.5c 119	. . . . . 6 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))
4	3	axL1.SH 30	. . . . 5 ⊢ (𝛾 → ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓))))
5	4	rcp-FR1.SH 40	. . . 4 ⊢ ((𝛾 → (𝜑 → 𝜓)) → (𝛾 → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓))))
6	2, 5	ax-MP 14	. . 3 ⊢ (𝛾 → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))
7	6	rcp-FR1.SH 40	. 2 ⊢ ((𝛾 → (𝜓 → 𝜑)) → (𝛾 → (𝜑 ↔ 𝜓)))
8	1, 7	ax-MP 14	1 ⊢ (𝛾 → (𝜑 ↔ 𝜓))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104

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

This theorem depends on definitions:  df-bi-D2.1 107

This theorem is referenced by:  bisym-P2.6b  124  subneg-P2.7  127  subiml-P2.8a  128  subimr-P2.8b  130  ndbii-P3.13  178