Theorem bicmb-P2.5c.2SH 120

Description: Inference from bicmb-P2.5c 119.

Hypotheses

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

Assertion

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

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

Step	Hyp	Ref	Expression
1		bicmb-P2.5c.2SH.2	. 2 ⊢ (𝜓 → 𝜑)
2		bicmb-P2.5c.2SH.1	. . 3 ⊢ (𝜑 → 𝜓)
3		bicmb-P2.5c 119	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))
4	2, 3	ax-MP 14	. 2 ⊢ ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓))
5	1, 4	ax-MP 14	1 ⊢ (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:  biref-P2.6a  123  truejust-P2.13  154