Theorem pm5.74ri 275

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		pm5.74 273	. 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
3	1, 2	mpbir 234	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  bitrd  282  bibi2d  345  tbt  372  cbvaldvaw  2071  sbiedvw  2132  sbiedw  2347  sbied  2533  sbco2d  2542  cbvraldva  3243  axgroth6  10906  isprm2  16850  ufileu  24231  bj-alnnf2  37620  qmapeldisjsim  39772