Theorem nfrexxy 2516

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2518 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 1466	. . 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 2511	. 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 1460   F/_wnfc 2306   E.wrex 2456

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461

This theorem is referenced by:  r19.12  2583  sbcrext  3040  nfuni  3815  nfiunxy  3912  rexxpf  4774  abrexex2g  6120  abrexex2  6124  nfrecs  6307  fimaxre2  11234  nfsum  11364  nfcprod1  11561  nfcprod  11562  bezoutlemmain  11998  ctiunctlemfo  12439  bj-findis  14701  strcollnfALT  14708