Theorem nfrexxy 2361

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2363 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 1355	. . 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 2357	. 2 |- ( T. -> F/ x E. y e. A ph )
7	6	trud 1252	1 |- F/ x E. y e. A ph

Syntax hints:   T. wtru 1244   F/wnf 1349   F/_wnfc 2165   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312

This theorem is referenced by:  r19.12  2422  sbcrext  2835  nfuni  3586  nfiunxy  3683  rexxpf  4483  abrexex2g  5747  abrexex2  5751  nfrecs  5922  bj-findis  10104  strcollnfALT  10111