Theorem ifelpwun 4578

Description: Existence of a conditional class, quantitative version (inference form). (Contributed by BJ, 15-Aug-2024.)

Hypotheses

Ref	Expression
ifelpwun.1	|- A e. _V
ifelpwun.2	|- B e. _V

Assertion

Ref	Expression
ifelpwun	|- if ( ph , A , B ) e. ~P ( A u. B )

Proof of Theorem ifelpwun

Step	Hyp	Ref	Expression
1		ifelpwun.1	. 2 |- A e. _V
2		ifelpwun.2	. 2 |- B e. _V
3		ifelpwung 4576	. 2 |- ( ( A e. _V /\ B e. _V ) -> if ( ph , A , B ) e. ~P ( A u. B ) )
4	1, 2, 3	mp2an 426	1 |- if ( ph , A , B ) e. ~P ( A u. B )

Syntax hints:   e. wcel 2200   _Vcvv 2800   u. cun 3196   ifcif 3603   ~Pcpw 3650

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-rab 2517  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-uni 3892

This theorem is referenced by:  fmelpw1o  7455  bj-charfun  16338