Theorem ifelpwund 4579

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 4578	. 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 2202   u. cun 3198   ifcif 3605   ~Pcpw 3652

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:  ifexd  4581