Theorem ifbid 2996

Description: Equivalence deduction for conditional operators.

Hypothesis

Ref	Expression
ifbid.1	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
ifbid	|- (ph -> if(ps, A, B) = if(ch, A, B))

Proof of Theorem ifbid

Step	Hyp	Ref	Expression
1		ifbid.1	. 2 |- (ph -> (ps <-> ch))
2		ifbi 2995	. 2 |- ((ps <-> ch) -> if(ps, A, B) = if(ch, A, B))
3	1, 2	syl 12	1 |- (ph -> if(ps, A, B) = if(ch, A, B))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298  ifcif 2982

This theorem is referenced by:  ifbieq12d 2998  oev 5198  unxpdomlem 5995  expval 7813  bcval 8210  ruclem4 8782  dscmet 9196  lmfexlem2 9235  gxval 9381  spwval2 9996  ifbieq2d 13594  txmet 15925  pcopt 16084  pcoass 16085

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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