Theorem eqif 2373

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 1481	. 2 |- (if(ph, B, C) = B -> (A = if(ph, B, C) <-> A = B))
2		eqeq2 1481	. 2 |- (if(ph, B, C) = C -> (A = if(ph, B, C) <-> A = C))
3	1, 2	elimif 2370	1 |- (A = if(ph, B, C) <-> ((ph /\ A = B) \/ (-. ph /\ A = C)))

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 954  ifcif 2357

This theorem is referenced by:  ifor 2377  dfrdg2 3924  dscmet 7870

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-if 2358

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