Theorem nfrexxy 2403

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2405 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 1395	. . 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 2399	. 2 |- ( T. -> F/ x E. y e. A ph )
7	6	trud 1293	1 |- F/ x E. y e. A ph

Syntax hints:   T. wtru 1285   F/wnf 1389   F/_wnfc 2206   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354

This theorem is referenced by:  r19.12  2466  sbcrext  2891  nfuni  3607  nfiunxy  3704  rexxpf  4501  abrexex2g  5767  abrexex2  5771  nfrecs  5945  fimaxre2  10109  bezoutlemmain  10387  bj-findis  10774  strcollnfALT  10781