Theorem ifeq1d 4485

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 4471	. 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 1537  ifcif 4467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-un 3941  df-if 4468

This theorem is referenced by:  ifeq12d  4487  ifbieq1d  4490  ifeq1da  4497  rabsnif  4659  fsuppmptif  8863  cantnflem1  9152  sumeq2w  15049  cbvsum  15052  isumless  15200  prodss  15301  subgmulg  18293  evlslem2  20292  selvval  20331  dmatcrng  21111  scmatscmiddistr  21117  scmatcrng  21130  marrepfval  21169  mdetr0  21214  mdetunilem8  21228  madufval  21246  madugsum  21252  minmar1fval  21255  decpmatid  21378  monmatcollpw  21387  pmatcollpwscmatlem1  21397  cnmpopc  23532  pcoval2  23620  pcopt  23626  itgz  24381  iblss2  24406  itgss  24412  itgcn  24443  plyeq0lem  24800  dgrcolem2  24864  plydivlem4  24885  leibpi  25520  chtublem  25787  sumdchr  25848  bposlem6  25865  lgsval  25877  dchrvmasumiflem2  26078  padicabvcxp  26208  dfrdg3  33041  matunitlindflem1  34903  ftc1anclem2  34983  ftc1anclem5  34986  ftc1anclem7  34988  hoidifhspval  42910  hoimbl  42933