Theorem nfrexw 2572

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2574 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 1515	. . 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 2567	. 2 |- ( T. -> F/ x E. y e. A ph )
7	6	mptru 1407	1 |- F/ x E. y e. A ph

Syntax hints:   T. wtru 1399   F/wnf 1509   F/_wnfc 2362   E.wrex 2512

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517

This theorem is referenced by:  r19.12  2640  sbcrext  3110  nfuni  3904  nfiunxy  4001  rexxpf  4883  abrexex2g  6291  abrexex2  6295  nfrecs  6516  nfwrd  11189  fimaxre2  11848  nfsum  11978  nfcprod1  12176  nfcprod  12177  bezoutlemmain  12630  ctiunctlemfo  13121  bj-findis  16675  strcollnfALT  16682