Theorem nfreudxy 2527

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 2219	. . . . . 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 2234	. . . 4 |- ( ph -> F/ x y e. A )
6		nfreudxy.3	. . . 4 |- ( ph -> F/ x ps )
7	5, 6	nfand 1500	. . 3 |- ( ph -> F/ x ( y e. A /\ ps ) )
8	1, 7	nfeud 1957	. 2 |- ( ph -> F/ x E! y ( y e. A /\ ps ) )
9		df-reu 2355	. . 3 |- ( E! y e. A ps <-> E! y ( y e. A /\ ps ) )
10	9	nfbii 1402	. 2 |- ( F/ x E! y e. A ps <-> F/ x E! y ( y e. A /\ ps ) )
11	8, 10	sylibr 132	1 |- ( ph -> F/ x E! y e. A ps )

Syntax hints:   -> wi 4   /\ wa 102   F/wnf 1389   e. wcel 1433   E!weu 1941   F/_wnfc 2206   E!wreu 2350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-cleq 2074  df-clel 2077  df-nfc 2208  df-reu 2355

This theorem is referenced by:  nfreuxy  2528