Theorem ifbieq2i 3495

Description: Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypotheses

Ref	Expression
ifbieq2i.1	⊢ (𝜑 ↔ 𝜓)
ifbieq2i.2	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
ifbieq2i	⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)

Proof of Theorem ifbieq2i

Step	Hyp	Ref	Expression
1		ifbieq2i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2		ifbi 3492	. . 3 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐴)
4		ifbieq2i.2	. . 3 ⊢ 𝐴 = 𝐵
5		ifeq2 3478	. . 3 ⊢ (𝐴 = 𝐵 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
6	4, 5	ax-mp 5	. 2 ⊢ if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)
7	3, 6	eqtri 2160	1 ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)

Syntax hints:   ↔ wb 104   = wceq 1331  ifcif 3474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-un 3075  df-if 3475

This theorem is referenced by:  ifbieq12i  3497  gcdcom  11662  gcdass  11703  lcmcom  11745  lcmass  11766