Theorem ifeq12d 3625

Description: Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)

Hypotheses

Ref	Expression
ifeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
ifeq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
ifeq12d	⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷))

Proof of Theorem ifeq12d

Step	Hyp	Ref	Expression
1		ifeq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	ifeq1d 3623	. 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3		ifeq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	ifeq2d 3624	. 2 ⊢ (𝜑 → if(𝜓, 𝐵, 𝐶) = if(𝜓, 𝐵, 𝐷))
5	2, 4	eqtrd 2264	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷))

Syntax hints:   → wi 4   = 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-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:  ifbieq12d  3632  xaddpnf1  10080  exp3val  10802  swrdccat3blem  11319  eucalgval  12625  ennnfonelemp1  13026  ennnfonelemnn0  13042  mulgfvalg  13707  mulgpropdg  13750  lgsval  15732