Theorem eqif 2381

Description: Expansion of an equality with a conditional operator.

Assertion

Ref	Expression
eqif	|- (A = if(ph, B, C) <-> ((ph /\ A = B) \/ (-. ph /\ A = C)))

Proof of Theorem eqif

Step	Hyp	Ref	Expression
1		eqeq2 1487	. 2 |- (if(ph, B, C) = B -> (A = if(ph, B, C) <-> A = B))
2		eqeq2 1487	. 2 |- (if(ph, B, C) = C -> (A = if(ph, B, C) <-> A = C))
3	1, 2	elimif 2378	1 |- (A = if(ph, B, C) <-> ((ph /\ A = B) \/ (-. ph /\ A = C)))

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 958  ifcif 2365

This theorem is referenced by:  ifor 2385  dfrdg2 3939  dscmet 7915

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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