Theorem ifexg 4606

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 4605	1 |- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2203   _Vcvv 2813   ifcif 3620

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-rab 2529  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915

This theorem is referenced by:  ifex  4607