Theorem pm5.32ri 644

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 643	. 2 ⊢ ((φ ⋀ ψ) ↔ (φ ⋀ χ))
3		ancom 435	. 2 ⊢ ((ψ ⋀ φ) ↔ (φ ⋀ ψ))
4		ancom 435	. 2 ⊢ ((χ ⋀ φ) ↔ (φ ⋀ χ))
5	2, 3, 4	3bitr4 183	1 ⊢ ((ψ ⋀ φ) ↔ (χ ⋀ φ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223

This theorem is referenced by:  pm5.36 649  2eu5 1446  rabsn 2435  dfoprab2 3976  th3qlem1 4298  xpsnen 4415  pw2en 4426  rankuni 4670  dfms2 7738  pjima 10015

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

Copyright terms: Public domain