Theorem ifeq2d 4485

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 4471	. 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 4466

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 2157  ax-12 2173  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 3496  df-un 3940  df-if 4467

This theorem is referenced by:  ifeq12d  4486  ifbieq2d  4491  ifeq2da  4497  ifcomnan  4520  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  42255  etransclem14  42527  etransclem17  42530  etransclem21  42534  etransclem25  42538  etransclem28  42541  etransclem31  42544  hsphoif  42852  hoidmvval  42853  hsphoival  42855  hoidmvlelem5  42875  hoidmvle  42876  ovnhoi  42879  hspmbllem2  42903