Theorem pm5.32rd 492

Description: Distribution of implication over biconditional (deduction rule).

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 491	. 2 ⊢ (φ → ((ψ ∧ χ) ↔ (ψ ∧ θ)))
3		ancom 333	. 2 ⊢ ((χ ∧ ψ) ↔ (ψ ∧ χ))
4		ancom 333	. 2 ⊢ ((θ ∧ ψ) ↔ (ψ ∧ θ))
5	2, 3, 4	3bitr4g 428	1 ⊢ (φ → ((χ ∧ ψ) ↔ (θ ∧ ψ)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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