Theorem pm5.32ri 490

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

Hypothesis

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

Assertion

Ref	Expression
pm5.32ri	⊢ ((ψ ∧ φ) ↔ (χ ∧ φ))

Proof of Theorem pm5.32ri

Step	Hyp	Ref	Expression
1		pm5.32i.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	pm5.32i 489	. 2 ⊢ ((φ ∧ ψ) ↔ (φ ∧ χ))
3		ancom 333	. 2 ⊢ ((ψ ∧ φ) ↔ (φ ∧ ψ))
4		ancom 333	. 2 ⊢ ((χ ∧ φ) ↔ (φ ∧ χ))
5	2, 3, 4	3bitr4 158	1 ⊢ ((ψ ∧ φ) ↔ (χ ∧ φ))

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

This theorem is referenced by:  2eu5 1071  dfoprab2 3021  th3qlem1 3250  xpsnen 3339  pw2en 3348

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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