Theorem ifexg 4582

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 4581	1 |- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   _Vcvv 2802   ifcif 3605

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894

This theorem is referenced by:  ifex  4583