Theorem pm5.74i 180

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

Hypothesis

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

Assertion

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

Proof of Theorem pm5.74i

Step	Hyp	Ref	Expression
1		pm5.74i.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		pm5.74 179	. 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
3	1, 2	mpbi 145	1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bitrd  188  imbi2i  226  bibi2d  232  ibib  245  ibibr  246  anclb  319  pm5.42  320  ancrb  322  equsalh  1774  equsal  1775  equsalv  1841  sb6a  2041  ralbiia  2547  dfdif3  3319  raaan  3602  snssb  3811  exmid01  4294  isprm4  12754