Theorem pm4.38 842

Description: Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm4.38	⊢ (((φ ↔ χ) ∧ (ψ ↔ θ)) → ((φ ∧ ψ) ↔ (χ ∧ θ)))

Proof of Theorem pm4.38

Step	Hyp	Ref	Expression
1		simpl 443	. 2 ⊢ (((φ ↔ χ) ∧ (ψ ↔ θ)) → (φ ↔ χ))
2		simpr 447	. 2 ⊢ (((φ ↔ χ) ∧ (ψ ↔ θ)) → (ψ ↔ θ))
3	1, 2	anbi12d 691	1 ⊢ (((φ ↔ χ) ∧ (ψ ↔ θ)) → ((φ ∧ ψ) ↔ (χ ∧ θ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358

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 177  df-an 360

This theorem is referenced by: (None)