Theorem ifbid 2376

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 2375	. 2 |- ((ps <-> ch) -> if(ps, A, B) = if(ch, A, B))
3	1, 2	syl 10	1 |- (ph -> if(ps, A, B) = if(ch, A, B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958  ifcif 2365

This theorem is referenced by:  oev 4159  unxpdomlem 4854  expvalt 6571  bcvalt 6958  ruclem4 7514  dscmet 7915  lmfexlem2 7954  spwval2 8649

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-if 2366

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