Theorem bicmb-P2.5c.2AC.2SH 122

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

Hypotheses

Ref	Expression
bicmb-P2.5c.2AC.2SH.1	⊢ (𝛾₁ → (𝛾₂ → (𝜑 → 𝜓)))
bicmb-P2.5c.2AC.2SH.2	⊢ (𝛾₁ → (𝛾₂ → (𝜓 → 𝜑)))

Assertion

Ref	Expression
bicmb-P2.5c.2AC.2SH	⊢ (𝛾₁ → (𝛾₂ → (𝜑 ↔ 𝜓)))

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

Step	Hyp	Ref	Expression
1		bicmb-P2.5c.2AC.2SH.2	. 2 ⊢ (𝛾₁ → (𝛾₂ → (𝜓 → 𝜑)))
2		bicmb-P2.5c.2AC.2SH.1	. . . 4 ⊢ (𝛾₁ → (𝛾₂ → (𝜑 → 𝜓)))
3		bicmb-P2.5c 119	. . . . . . 7 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))
4	3	axL1.SH 30	. . . . . 6 ⊢ (𝛾₂ → ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓))))
5	4	axL1.SH 30	. . . . 5 ⊢ (𝛾₁ → (𝛾₂ → ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))))
6	5	rcp-FR2.SH 42	. . . 4 ⊢ ((𝛾₁ → (𝛾₂ → (𝜑 → 𝜓))) → (𝛾₁ → (𝛾₂ → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))))
7	2, 6	ax-MP 14	. . 3 ⊢ (𝛾₁ → (𝛾₂ → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓))))
8	7	rcp-FR2.SH 42	. 2 ⊢ ((𝛾₁ → (𝛾₂ → (𝜓 → 𝜑))) → (𝛾₁ → (𝛾₂ → (𝜑 ↔ 𝜓))))
9	1, 8	ax-MP 14	1 ⊢ (𝛾₁ → (𝛾₂ → (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:  bitrns-P2.6c  126