Theorem ifelpwund 4572

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

Hypotheses

Ref	Expression
ifelpwund.1	|- ( ph -> A e. V )
ifelpwund.2	|- ( ph -> B e. W )

Assertion

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

Proof of Theorem ifelpwund

Step	Hyp	Ref	Expression
1		ifelpwund.1	. 2 |- ( ph -> A e. V )
2		ifelpwund.2	. 2 |- ( ph -> B e. W )
3		ifelpwung 4571	. 2 |- ( ( A e. V /\ B e. W ) -> if ( ps , A , B ) e. ~P ( A u. B ) )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> if ( ps , A , B ) e. ~P ( A u. B ) )

Syntax hints:   -> wi 4   e. wcel 2200   u. cun 3195   ifcif 3602   ~Pcpw 3649

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 4201  ax-pr 4292  ax-un 4523

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888

This theorem is referenced by:  ifexd  4574