Theorem ifbieq2i 3629

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 3626	. . 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 3609	. . 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 2252	1 ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)

Syntax hints:   ↔ wb 105   = wceq 1397  ifcif 3605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-un 3204  df-if 3606

This theorem is referenced by:  ifbieq12i  3631  gcdcom  12543  gcdass  12585  lcmcom  12635  lcmass  12656