Theorem imbi1 347

Description: Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
imbi1	⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒)))

Proof of Theorem imbi1

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓))
2	1	imbi1d 341	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒)))

Syntax hints:   → wi 4   ↔ wb 206

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 207

This theorem is referenced by:  imbi1i  349  nanbi1  1501  ifpbi1  43468  3impexpVD  44847  ancomstVD  44856  onfrALTVD  44882  hbimpgVD  44895  hbexgVD  44897  ax6e2ndeqVD  44900  ax6e2ndeqALT  44922