Theorem ifelpwund 4585

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 4584	. 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 3199   ifcif 3607   ~Pcpw 3656

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-rab 2520  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899

This theorem is referenced by:  ifexd  4587