Theorem nfrexxy 2514

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2516 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)

Hypotheses

Ref	Expression
nfralxy.1	|- F/_ x A
nfralxy.2	|- F/ x ph

Assertion

Ref	Expression
nfrexxy	|- F/ x E. y e. A ph

Distinct variable group:   x, y

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

Proof of Theorem nfrexxy

Step	Hyp	Ref	Expression
1		nftru 1464	. . 3 |- F/ y T.
2		nfralxy.1	. . . 4 |- F/_ x A
3	2	a1i 9	. . 3 |- ( T. -> F/_ x A )
4		nfralxy.2	. . . 4 |- F/ x ph
5	4	a1i 9	. . 3 |- ( T. -> F/ x ph )
6	1, 3, 5	nfrexdxy 2509	. 2 |- ( T. -> F/ x E. y e. A ph )
7	6	mptru 1362	1 |- F/ x E. y e. A ph

Syntax hints:   T. wtru 1354   F/wnf 1458   F/_wnfc 2304   E.wrex 2454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-4 1508  ax-17 1524  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459

This theorem is referenced by:  r19.12  2581  sbcrext  3038  nfuni  3811  nfiunxy  3908  rexxpf  4767  abrexex2g  6111  abrexex2  6115  nfrecs  6298  fimaxre2  11203  nfsum  11333  nfcprod1  11530  nfcprod  11531  bezoutlemmain  11966  ctiunctlemfo  12407  bj-findis  14300  strcollnfALT  14307