Theorem anbi1 634

Description: Introduce a right conjunct to both sides of a logical equivalence. Theorem *4.36 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
anbi1	⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒)))

Proof of Theorem anbi1

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓))
2	1	anbi1d 632	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399

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 210  df-an 400

This theorem is referenced by:  pm5.75  1026  relexpindlem  14414  rexfiuz  14699  bnj916  32315  redundpim3  36025