Theorem ifbieq12i 3595

Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)

Hypotheses

Ref	Expression
ifbieq12i.1	⊢ (𝜑 ↔ 𝜓)
ifbieq12i.2	⊢ 𝐴 = 𝐶
ifbieq12i.3	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
ifbieq12i	⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)

Proof of Theorem ifbieq12i

Step	Hyp	Ref	Expression
1		ifbieq12i.2	. . 3 ⊢ 𝐴 = 𝐶
2		ifeq1 3573	. . 3 ⊢ (𝐴 = 𝐶 → if(𝜑, 𝐴, 𝐵) = if(𝜑, 𝐶, 𝐵))
3	1, 2	ax-mp 5	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜑, 𝐶, 𝐵)
4		ifbieq12i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
5		ifbieq12i.3	. . 3 ⊢ 𝐵 = 𝐷
6	4, 5	ifbieq2i 3593	. 2 ⊢ if(𝜑, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷)
7	3, 6	eqtri 2225	1 ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)

Syntax hints:   ↔ wb 105   = wceq 1372  ifcif 3570

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rab 2492  df-v 2773  df-un 3169  df-if 3571

This theorem is referenced by: (None)