Theorem ifbieq2i 3559

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 3556	. . 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 3540	. . 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 2198	1 ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)

Syntax hints:   ↔ wb 105   = wceq 1353  ifcif 3536

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-un 3135  df-if 3537

This theorem is referenced by:  ifbieq12i  3561  gcdcom  11977  gcdass  12019  lcmcom  12067  lcmass  12088