Theorem nfrexw 2546

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2548 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 1490	. . 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 2541	. 2 |- ( T. -> F/ x E. y e. A ph )
7	6	mptru 1382	1 |- F/ x E. y e. A ph

Syntax hints:   T. wtru 1374   F/wnf 1484   F/_wnfc 2336   E.wrex 2486

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491

This theorem is referenced by:  r19.12  2613  sbcrext  3080  nfuni  3861  nfiunxy  3958  rexxpf  4832  abrexex2g  6217  abrexex2  6221  nfrecs  6405  nfwrd  11039  fimaxre2  11608  nfsum  11738  nfcprod1  11935  nfcprod  11936  bezoutlemmain  12389  ctiunctlemfo  12880  bj-findis  16049  strcollnfALT  16056