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  1737  equsal  1738  equsalv  1804  sb6a  2004  ralbiia  2508  dfdif3  3269  raaan  3552  snssb  3751  exmid01  4227  isprm4  12257