Theorem ifeq1d 4054

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 4040	. 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 1475  ifcif 4036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-un 3545  df-if 4037

This theorem is referenced by:  ifeq12d  4056  ifbieq1d  4059  ifeq1da  4066  rabsnif  4202  fsuppmptif  8188  cantnflem1  8469  sumeq2w  14270  cbvsum  14273  isumless  14416  prodss  14516  subgmulg  17431  evlslem2  19333  dmatcrng  20127  scmatscmiddistr  20133  scmatcrng  20146  marrepfval  20185  mdetr0  20230  mdetunilem8  20244  madufval  20262  madugsum  20268  minmar1fval  20271  decpmatid  20394  monmatcollpw  20403  pmatcollpwscmatlem1  20413  cnmpt2pc  22535  pcoval2  22624  pcopt  22630  itgz  23353  iblss2  23378  itgss  23384  itgcn  23415  plyeq0lem  23770  dgrcolem2  23834  plydivlem4  23855  leibpi  24469  chtublem  24736  sumdchr  24797  bposlem6  24814  lgsval  24826  dchrvmasumiflem2  24991  padicabvcxp  25121  dfrdg3  30946  matunitlindflem1  32575  ftc1anclem2  32656  ftc1anclem5  32659  ftc1anclem7  32661  hoidifhspval  39498  hoimbl  39521