Theorem pm5.74ri 271

Description: Distribution of implication over biconditional (reverse inference form). (Contributed by NM, 1-Aug-1994.)

Hypothesis

Ref	Expression
pm5.74ri.1	⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))

Assertion

Ref	Expression
pm5.74ri	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Proof of Theorem pm5.74ri

Step	Hyp	Ref	Expression
1		pm5.74ri.1	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))
2		pm5.74 269	. 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
3	1, 2	mpbir 230	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206

This theorem is referenced by:  bitrd  278  bibi2d  341  tbt  368  cbvaldvaw  2039  sbiedvw  2094  sbiedw  2307  sbied  2500  sbco2d  2509  2mos  2643  cbvraldva  3234  cbvraldva2  3342  axgroth6  10825  isprm2  16623  ufileu  23643