Theorem pm5.32rd 424

Description: Distribution of implication over biconditional (deduction rule). (Contributed by NM, 25-Dec-2004.)

Hypothesis

Ref	Expression
pm5.32d.1	⊢ (φ → (ψ → (χ ↔ θ)))

Assertion

Ref	Expression
pm5.32rd	⊢ (φ → ((χ ∧ ψ) ↔ (θ ∧ ψ)))

Proof of Theorem pm5.32rd

Step	Hyp	Ref	Expression
1		pm5.32d.1	. . 3 ⊢ (φ → (ψ → (χ ↔ θ)))
2	1	pm5.32d 423	. 2 ⊢ (φ → ((ψ ∧ χ) ↔ (ψ ∧ θ)))
3		ancom 253	. 2 ⊢ ((χ ∧ ψ) ↔ (ψ ∧ χ))
4		ancom 253	. 2 ⊢ ((θ ∧ ψ) ↔ (ψ ∧ θ))
5	2, 3, 4	3bitr4g 212	1 ⊢ (φ → ((χ ∧ ψ) ↔ (θ ∧ ψ)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  anbi1d  438  pm5.71dc  867  1idprl  6566  1idpru  6567