Theorem ifex 2404

Description: Conditional operator existence.

Hypotheses

Ref	Expression
dedex.1	|- A e. V
dedex.2	|- B e. V

Assertion

Ref	Expression
ifex	|- if(ph, A, B) e. V

Proof of Theorem ifex

Step	Hyp	Ref	Expression
1		dedex.1	. 2 |- A e. V
2		dedex.2	. 2 |- B e. V
3	1, 2	keepel 2403	1 |- if(ph, A, B) e. V

Syntax hints:   e. wcel 960  Vcvv 1814  ifcif 2365

This theorem is referenced by:  oev 4159  unxpdomlem 4854  expvalt 6571  bcvalt 6958  climuni 7099  cvgcmp3cetlem2 7189  ruclem15 7525  hlimcau 9102  hlimuni 9104

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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