Theorem anbi1 687

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 19	. 2 ⊢ ((φ ↔ ψ) → (φ ↔ ψ))
2	1	anbi1d 685	1 ⊢ ((φ ↔ ψ) → ((φ ∧ χ) ↔ (ψ ∧ χ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360

This theorem is used by:  pm5.75  903  nanbi1  1295