Theorem ifeq2d 2994

Description: Equality deduction for conditional operator.

Hypothesis

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

Assertion

Ref	Expression
ifeq2d	|- (ph -> if(ps, C, A) = if(ps, C, B))

Proof of Theorem ifeq2d

Step	Hyp	Ref	Expression
1		ifeq1d.1	. 2 |- (ph -> A = B)
2		ifeq2 2987	. 2 |- (A = B -> if(ps, C, A) = if(ps, C, B))
3	1, 2	syl 12	1 |- (ph -> if(ps, C, A) = if(ps, C, B))

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

This theorem is referenced by:  ifbieq12d 2998  rdgeq1 5142  oev 5198  unxpdomlem 5995  expval 7813  gxoprval 9380  gxval 9381  spwval2 9996  ifbieq2d 13594  prodeq2 14661  ifeq2da 15694  phtpycolem1 16051  phtpycolem2 16052  pcoval 16073  pcohtpylem1 16080  pcohtpylem2 16081  pcopt 16084  pcorev 16087

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

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