Theorem pm5.32ri 459

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 458	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))
3		ancom 266	. 2 ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜑 ∧ 𝜓))
4		ancom 266	. 2 ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜑 ∧ 𝜒))
5	2, 3, 4	3bitr4i 212	1 ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜒 ∧ 𝜑))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  anbi1i  462  pm5.36  618  pm5.61  806  oranabs  827  ceqsralt  2849  ceqsrexbv  2957  reuind  3031  rabsn  3776  dfoprab2  6138  xpsnen  7124  elfpw  7267  sspw1or2  7551  nn1suc  9333  isprm2  12921  ismnd  13792  dfgrp2e  13893  isxms2  15690  clwwlkn1  16876  clwwlkn2  16879  2alsraln0m  17377