Theorem bimpr-P4 533

Description: Modus Ponens with '↔' (reverse). †

Hypotheses

Ref	Expression
bimpr-P4.1	⊢ (𝛾 → 𝜓)
bimpr-P4.2	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bimpr-P4	⊢ (𝛾 → 𝜑)

Proof of Theorem bimpr-P4

Step	Hyp	Ref	Expression
1		bimpr-P4.1	. 2 ⊢ (𝛾 → 𝜓)
2		bimpr-P4.2	. . 3 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
3	2	ndbier-P3.15 180	. 2 ⊢ (𝛾 → (𝜓 → 𝜑))
4	1, 3	ndime-P3.6 171	1 ⊢ (𝛾 → 𝜑)

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:  bimpr-P4.RC  534  eqmiddle-P6  708  example-E7.1a  1074