Theorem nfreudxy 2643

Description: Not-free deduction for restricted uniqueness. This is a version where x and y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)

Hypotheses

Ref	Expression
nfreudxy.1	|- F/ y ph
nfreudxy.2	|- ( ph -> F/_ x A )
nfreudxy.3	|- ( ph -> F/ x ps )

Assertion

Ref	Expression
nfreudxy	|- ( ph -> F/ x E! y e. A ps )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   A( x, y)

Proof of Theorem nfreudxy

Step	Hyp	Ref	Expression
1		nfreudxy.1	. . 3 |- F/ y ph
2		nfcv 2312	. . . . . 6 |- F/_ x y
3	2	a1i 9	. . . . 5 |- ( ph -> F/_ x y )
4		nfreudxy.2	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeld 2328	. . . 4 |- ( ph -> F/ x y e. A )
6		nfreudxy.3	. . . 4 |- ( ph -> F/ x ps )
7	5, 6	nfand 1561	. . 3 |- ( ph -> F/ x ( y e. A /\ ps ) )
8	1, 7	nfeud 2035	. 2 |- ( ph -> F/ x E! y ( y e. A /\ ps ) )
9		df-reu 2455	. . 3 |- ( E! y e. A ps <-> E! y ( y e. A /\ ps ) )
10	9	nfbii 1466	. 2 |- ( F/ x E! y e. A ps <-> F/ x E! y ( y e. A /\ ps ) )
11	8, 10	sylibr 133	1 |- ( ph -> F/ x E! y e. A ps )

Syntax hints:   -> wi 4   /\ wa 103   F/wnf 1453   E!weu 2019   e. wcel 2141   F/_wnfc 2299   E!wreu 2450

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-cleq 2163  df-clel 2166  df-nfc 2301  df-reu 2455

This theorem is referenced by:  nfreuxy  2644