Theorem pm5.32rd 621

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 620	. 2 ⊢ (φ → ((ψ ∧ χ) ↔ (ψ ∧ θ)))
3		ancom 437	. 2 ⊢ ((χ ∧ ψ) ↔ (ψ ∧ χ))
4		ancom 437	. 2 ⊢ ((θ ∧ ψ) ↔ (ψ ∧ θ))
5	2, 3, 4	3bitr4g 279	1 ⊢ (φ → ((χ ∧ ψ) ↔ (θ ∧ ψ)))

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:  anbi1d  685  pm5.71  902