Theorem ifelpwund 4457

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 ( ph , 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 4456	. 2 |- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. ~P ( A u. B ) )
4	1, 2, 3	syl2anc 409	1 |- ( ph -> if ( ph , A , B ) e. ~P ( A u. B ) )

Syntax hints:   -> wi 4   e. wcel 2135   u. cun 3112   ifcif 3518   ~Pcpw 3556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pr 4184  ax-un 4408

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-rab 2451  df-v 2726  df-un 3118  df-in 3120  df-ss 3127  df-if 3519  df-pw 3558  df-sn 3579  df-pr 3580  df-uni 3787

This theorem is referenced by:  ifexd  4459