Theorem nfrexxy 2472

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2474 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 1442	. . 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 2468	. 2 |- ( T. -> F/ x E. y e. A ph )
7	6	mptru 1340	1 |- F/ x E. y e. A ph

Syntax hints:   T. wtru 1332   F/wnf 1436   F/_wnfc 2268   E.wrex 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422

This theorem is referenced by:  r19.12  2538  sbcrext  2986  nfuni  3742  nfiunxy  3839  rexxpf  4686  abrexex2g  6018  abrexex2  6022  nfrecs  6204  fimaxre2  10998  nfsum  11126  nfcprod1  11323  nfcprod  11324  bezoutlemmain  11686  ctiunctlemfo  11952  bj-findis  13177  strcollnfALT  13184