Theorem ifeq1d 2993

Description: Equality deduction for conditional operator.

Hypothesis

Ref	Expression
ifeq1d.1	|- (ph -> A = B)

Assertion

Ref	Expression
ifeq1d	|- (ph -> if(ps, A, C) = if(ps, B, C))

Proof of Theorem ifeq1d

Step	Hyp	Ref	Expression
1		ifeq1d.1	. 2 |- (ph -> A = B)
2		ifeq1 2985	. 2 |- (A = B -> if(ps, A, C) = if(ps, B, C))
3	1, 2	syl 12	1 |- (ph -> if(ps, A, C) = if(ps, B, C))

Syntax hints:   -> wi 3   = wceq 1298  ifcif 2982

This theorem is referenced by:  ifbieq12d 2998  oev 5198  bcval 8210  ruclem4 8782  lmfexlem2 9235  gxoprval 9380  gxval 9381  spwval2 9996  ifeq1da 15693  phtpycolem1 16051  phtpycolem2 16052  pcoval 16073  pcohtpylem1 16080  pcohtpylem2 16081

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-if 2983

Copyright terms: Public domain