Theorem imbi2 348

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 340	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 → 𝜑) ↔ (𝜒 → 𝜓)))

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:  imbibi  391  con3ALT  1085  relexpindlem  15112  relexpind  15113  unielss  43179  ifpbi2  43429  ifpbi3  43430  3impexpbicom  44450  sbcim2g  44509  3impexpbicomVD  44828  sbcim2gVD  44846  csbeq2gVD  44863  con5VD  44871  hbexgVD  44877  ax6e2ndeqVD  44880  2sb5ndVD  44881  ax6e2ndeqALT  44902  2sb5ndALT  44903