Theorem ifeq12d 3545

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 3543	. 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3		ifeq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	ifeq2d 3544	. 2 ⊢ (𝜑 → if(𝜓, 𝐵, 𝐶) = if(𝜓, 𝐵, 𝐷))
5	2, 4	eqtrd 2203	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷))

Syntax hints:   → wi 4   = wceq 1348  ifcif 3526

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-un 3125  df-if 3527

This theorem is referenced by:  ifbieq12d  3552  xaddpnf1  9803  exp3val  10478  eucalgval  12008  ennnfonelemp1  12361  ennnfonelemnn0  12377  lgsval  13699