Theorem imbi2 337

Description: Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)

Assertion

Ref	Expression
imbi2	⊢ ((𝜑 ↔ 𝜓) → ((𝜒 → 𝜑) ↔ (𝜒 → 𝜓)))

Proof of Theorem imbi2

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓))
2	1	imbi2d 329	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 → 𝜑) ↔ (𝜒 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 196

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 197

This theorem is referenced by:  relexpindlem  14007  relexpind  14008  ifpbi2  38334  ifpbi3  38335  3impexpbicom  39207  sbcim2g  39270  3impexpbicomVD  39611  sbcim2gVD  39630  csbeq2gVD  39647  con5VD  39655  hbexgVD  39661  ax6e2ndeqVD  39664  2sb5ndVD  39665  ax6e2ndeqALT  39686  2sb5ndALT  39687