Theorem nfunidALT2 34256

Description: Deduction version of nfuni 4442. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nfunidALT2.1	|- ( ph -> F/_ x A )

Assertion

Ref	Expression
nfunidALT2	|- ( ph -> F/_ x U. A )

Proof of Theorem nfunidALT2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfaba1 2770	. . 3 |- F/_ x { y | A. x y e. A }
2	1	nfuni 4442	. 2 |- F/_ x U. { y | A. x y e. A }
3		nfunidALT2.1	. . 3 |- ( ph -> F/_ x A )
4		nfnfc1 2767	. . . 4 |- F/ x F/_ x A
5		abidnf 3375	. . . . 5 |- ( F/_ x A -> { y | A. x y e. A } = A )
6	5	unieqd 4446	. . . 4 |- ( F/_ x A -> U. { y | A. x y e. A } = U. A )
7	4, 6	nfceqdf 2760	. . 3 |- ( F/_ x A -> ( F/_ x U. { y | A. x y e. A } <-> F/_ x U. A ) )
8	3, 7	syl 17	. 2 |- ( ph -> ( F/_ x U. { y | A. x y e. A } <-> F/_ x U. A ) )
9	2, 8	mpbii 223	1 |- ( ph -> F/_ x U. A )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   {cab 2608   F/_wnfc 2751   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-uni 4437

This theorem is referenced by: (None)