Theorem nfralxy 2414

Description: Not-free for restricted universal quantification where x and y are distinct. See nfralya 2416 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
nfralxy	|- F/ x A. y e. A ph

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   A( x, y)

Proof of Theorem nfralxy

Step	Hyp	Ref	Expression
1		nftru 1400	. . 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	nfraldxy 2410	. 2 |- ( T. -> F/ x A. y e. A ph )
7	6	mptru 1298	1 |- F/ x A. y e. A ph

Syntax hints:   T. wtru 1290   F/wnf 1394   F/_wnfc 2215   A.wral 2359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364

This theorem is referenced by:  nfra2xy  2418  rspc2  2731  sbcralt  2913  sbcralg  2915  raaanlem  3383  nfint  3693  nfiinxy  3752  nfpo  4119  nfso  4120  nfse  4159  nffrfor  4166  nfwe  4173  ralxpf  4570  funimaexglem  5083  fun11iun  5258  dff13f  5531  nfiso  5567  mpt2eq123  5690  nfofr  5844  fmpt2x  5952  nfrecs  6054  xpf1o  6540  ac6sfi  6594  lble  8380  fzrevral  9486  nfsum1  10709  nfsum  10710  fsum2dlemstep  10791  fisumcom2  10795  bezoutlemmain  11069  setindis  11508  bdsetindis  11510