Theorem pm5.32ri 451

Description: Distribution of implication over biconditional (inference form). (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 450	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))
3		ancom 264	. 2 ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜑 ∧ 𝜓))
4		ancom 264	. 2 ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜑 ∧ 𝜒))
5	2, 3, 4	3bitr4i 211	1 ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜒 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  anbi1i  454  pm5.36  600  pm5.61  784  oranabs  805  ceqsralt  2753  ceqsrexbv  2857  reuind  2931  rabsn  3643  dfoprab2  5889  xpsnen  6787  nn1suc  8876  isprm2  12049  isxms2  13092