Theorem ifeq1d 4324

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

Hypothesis

Ref	Expression
ifeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
ifeq1d	⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Proof of Theorem ifeq1d

Step	Hyp	Ref	Expression
1		ifeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		ifeq1 4310	. 2 ⊢ (𝐴 = 𝐵 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3	1, 2	syl 17	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Syntax hints:   → wi 4   = wceq 1658  ifcif 4306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-un 3803  df-if 4307

This theorem is referenced by:  ifeq12d  4326  ifbieq1d  4329  ifeq1da  4336  rabsnif  4476  fsuppmptif  8574  cantnflem1  8863  sumeq2w  14799  cbvsum  14802  isumless  14951  prodss  15050  subgmulg  17959  evlslem2  19872  dmatcrng  20676  scmatscmiddistr  20682  scmatcrng  20695  marrepfval  20734  mdetr0  20779  mdetunilem8  20793  madufval  20811  madugsum  20817  minmar1fval  20820  decpmatid  20945  monmatcollpw  20954  pmatcollpwscmatlem1  20964  cnmpt2pc  23097  pcoval2  23185  pcopt  23191  itgz  23946  iblss2  23971  itgss  23977  itgcn  24008  plyeq0lem  24365  dgrcolem2  24429  plydivlem4  24450  leibpi  25082  chtublem  25349  sumdchr  25410  bposlem6  25427  lgsval  25439  dchrvmasumiflem2  25604  padicabvcxp  25734  dfrdg3  32240  matunitlindflem1  33949  ftc1anclem2  34029  ftc1anclem5  34032  ftc1anclem7  34034  hoidifhspval  41616  hoimbl  41639