Theorem ifeq2d 4486

Description: Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)

Hypothesis

Ref	Expression
ifeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
ifeq2d	⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Proof of Theorem ifeq2d

Step	Hyp	Ref	Expression
1		ifeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		ifeq2 4472	. 2 ⊢ (𝐴 = 𝐵 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
3	1, 2	syl 17	1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Syntax hints:   → wi 4   = wceq 1533  ifcif 4467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-un 3941  df-if 4468

This theorem is referenced by:  ifeq12d  4487  ifbieq2d  4492  ifeq2da  4498  ifcomnan  4521  rdgeq1  8041  cantnflem1d  9145  cantnflem1  9146  rexmul  12658  1arithlem4  16256  ramcl  16359  mplcoe1  20240  mplcoe5  20243  subrgascl  20272  selvffval  20323  selvval  20325  scmatscm  21116  marrepfval  21163  ma1repveval  21174  mulmarep1el  21175  mdetralt2  21212  mdetunilem8  21222  maduval  21241  maducoeval2  21243  madurid  21247  minmar1val0  21250  monmatcollpw  21381  pmatcollpwscmatlem1  21391  monmat2matmon  21426  itg2monolem1  24345  iblmulc2  24425  itgmulc2lem1  24426  bddmulibl  24433  dvtaylp  24952  dchrinvcl  25823  rpvmasum2  26082  padicfval  26186  plymulx  31813  itg2addnclem  34937  itg2addnclem3  34939  itg2addnc  34940  itgmulc2nclem1  34952  hdmap1fval  38926  itgioocnicc  42254  etransclem14  42526  etransclem17  42529  etransclem21  42533  etransclem25  42537  etransclem28  42540  etransclem31  42543  hsphoif  42851  hoidmvval  42852  hsphoival  42854  hoidmvlelem5  42874  hoidmvle  42875  ovnhoi  42878  hspmbllem2  42902