Theorem ifelpwund 4517

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 4516	. 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 2167   u. cun 3155   ifcif 3561   ~Pcpw 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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840

This theorem is referenced by:  ifexd  4519