Theorem nfrexw 2569

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2571 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 1512	. . 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 2564	. 2 |- ( T. -> F/ x E. y e. A ph )
7	6	mptru 1404	1 |- F/ x E. y e. A ph

Syntax hints:   T. wtru 1396   F/wnf 1506   F/_wnfc 2359   E.wrex 2509

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514

This theorem is referenced by:  r19.12  2637  sbcrext  3106  nfuni  3893  nfiunxy  3990  rexxpf  4866  abrexex2g  6255  abrexex2  6259  nfrecs  6443  nfwrd  11086  fimaxre2  11724  nfsum  11854  nfcprod1  12051  nfcprod  12052  bezoutlemmain  12505  ctiunctlemfo  12996  bj-findis  16272  strcollnfALT  16279