Theorem imbi2 338

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 330	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  13803  relexpind  13804  ifpbi2  37811  ifpbi3  37812  3impexpbicom  38685  sbcim2g  38748  3impexpbicomVD  39092  sbcim2gVD  39111  csbeq2gVD  39128  con5VD  39136  hbexgVD  39142  ax6e2ndeqVD  39145  2sb5ndVD  39146  ax6e2ndeqALT  39167  2sb5ndALT  39168