Theorem elunif 39175

Description: A version of eluni 4439 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
elunif.1	|- F/_ x A
elunif.2	|- F/_ x B

Assertion

Ref	Expression
elunif	|- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )

Distinct variable group:   A, B

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem elunif

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4439	. 2 |- ( A e. U. B <-> E. y ( A e. y /\ y e. B ) )
2		elunif.1	. . . . 5 |- F/_ x A
3		nfcv 2764	. . . . 5 |- F/_ x y
4	2, 3	nfel 2777	. . . 4 |- F/ x A e. y
5		elunif.2	. . . . 5 |- F/_ x B
6	3, 5	nfel 2777	. . . 4 |- F/ x y e. B
7	4, 6	nfan 1828	. . 3 |- F/ x ( A e. y /\ y e. B )
8		nfv 1843	. . 3 |- F/ y ( A e. x /\ x e. B )
9		eleq2 2690	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
10		eleq1 2689	. . . 4 |- ( y = x -> ( y e. B <-> x e. B ) )
11	9, 10	anbi12d 747	. . 3 |- ( y = x -> ( ( A e. y /\ y e. B ) <-> ( A e. x /\ x e. B ) ) )
12	7, 8, 11	cbvex 2272	. 2 |- ( E. y ( A e. y /\ y e. B ) <-> E. x ( A e. x /\ x e. B ) )
13	1, 12	bitri 264	1 |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   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-v 3202  df-uni 4437

This theorem is referenced by:  eluni2f  39286  stoweidlem46  40263  stoweidlem57  40274