Theorem pm5.32ri 619

Description: Distribution of implication over biconditional (inference rule). (Contributed by NM, 12-Mar-1995.)

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 618	. 2 ⊢ ((φ ∧ ψ) ↔ (φ ∧ χ))
3		ancom 437	. 2 ⊢ ((ψ ∧ φ) ↔ (φ ∧ ψ))
4		ancom 437	. 2 ⊢ ((χ ∧ φ) ↔ (φ ∧ χ))
5	2, 3, 4	3bitr4i 268	1 ⊢ ((ψ ∧ φ) ↔ (χ ∧ φ))

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:  anbi1i  676  pm5.61  693  oranabs  829  pm5.36  849  2eu5  2288  ceqsralt  2882  ceqsrexbv  2973  reuind  3039  rabsn  3790  elpw1  4144  pw1in  4164  addccan2nclem1  6263  scancan  6331