Theorem nfrexw 2536

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2538 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 1480	. . 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 2531	. 2 |- ( T. -> F/ x E. y e. A ph )
7	6	mptru 1373	1 |- F/ x E. y e. A ph

Syntax hints:   T. wtru 1365   F/wnf 1474   F/_wnfc 2326   E.wrex 2476

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481

This theorem is referenced by:  r19.12  2603  sbcrext  3067  nfuni  3846  nfiunxy  3943  rexxpf  4814  abrexex2g  6178  abrexex2  6182  nfrecs  6366  nfwrd  10965  fimaxre2  11394  nfsum  11524  nfcprod1  11721  nfcprod  11722  bezoutlemmain  12175  ctiunctlemfo  12666  bj-findis  15635  strcollnfALT  15642