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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  bitrd  188  imbi2i  226  bibi2d  232  ibib  245  ibibr  246  anclb  319  pm5.42  320  ancrb  322  equsalh  1778  equsal  1779  equsalv  1846  sb6a  2048  ralbiia  2564  dfdif3  3339  raaan  3633  snssb  3848  exmid01  4335  isprm4  12897