Theorem nfrexw 2571

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2573 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
nfrexw	|- F/ x E. y e. A ph

Distinct variable group:   x, y

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

Proof of Theorem nfrexw

Step	Hyp	Ref	Expression
1		nftru 1514	. . 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 2566	. 2 |- ( T. -> F/ x E. y e. A ph )
7	6	mptru 1406	1 |- F/ x E. y e. A ph

Syntax hints:   T. wtru 1398   F/wnf 1508   F/_wnfc 2361   E.wrex 2511

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516

This theorem is referenced by:  r19.12  2639  sbcrext  3109  nfuni  3899  nfiunxy  3996  rexxpf  4877  abrexex2g  6282  abrexex2  6286  nfrecs  6473  nfwrd  11146  fimaxre2  11792  nfsum  11922  nfcprod1  12120  nfcprod  12121  bezoutlemmain  12574  ctiunctlemfo  13065  bj-findis  16600  strcollnfALT  16607