Theorem nfrexw 2589

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2591 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 1519	. . 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 2584	. 2 |- ( T. -> F/ x E. y e. A ph )
7	6	mptru 1411	1 |- F/ x E. y e. A ph

Syntax hints:   T. wtru 1403   F/wnf 1513   F/_wnfc 2379   E.wrex 2529

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534

This theorem is referenced by:  r19.12  2657  sbcrext  3129  nfuni  3939  nfiunxy  4036  rexxpf  4925  abrexex2g  6343  abrexex2  6347  nfrecs  6572  nfwrd  11316  fimaxre2  11976  nfsum  12106  nfcprod1  12304  nfcprod  12305  bezoutlemmain  12758  ctiunctlemfo  13313  bj-findis  16988  strcollnfALT  16995  nfrals  17119