Theorem ifexg 4626

Description: Existence of the conditional operator (closed form). (Contributed by NM, 21-Mar-2011.) (Proof shortened by BJ, 1-Sep-2022.)

Assertion

Ref	Expression
ifexg	|- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. _V )

Proof of Theorem ifexg

Step	Hyp	Ref	Expression
1		simpl 109	. 2 |- ( ( A e. V /\ B e. W ) -> A e. V )
2		simpr 110	. 2 |- ( ( A e. V /\ B e. W ) -> B e. W )
3	1, 2	ifexd 4625	1 |- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2209   _Vcvv 2821   ifcif 3635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931

This theorem is referenced by:  ifex  4627