Theorem nfralxy 2474

Description: Not-free for restricted universal quantification where x and y are distinct. See nfralya 2476 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 1443	. . 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 2470	. 2 |- ( T. -> F/ x A. y e. A ph )
7	6	mptru 1341	1 |- F/ x A. y e. A ph

Syntax hints:   T. wtru 1333   F/wnf 1437   F/_wnfc 2269   A.wral 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422

This theorem is referenced by:  nfra2xy  2478  rspc2  2804  sbcralt  2989  sbcralg  2991  raaanlem  3473  nfint  3789  nfiinxy  3848  nfpo  4231  nfso  4232  nfse  4271  nffrfor  4278  nfwe  4285  ralxpf  4693  funimaexglem  5214  fun11iun  5396  dff13f  5679  nfiso  5715  mpoeq123  5838  nfofr  5996  fmpox  6106  nfrecs  6212  xpf1o  6746  ac6sfi  6800  ismkvnex  7037  lble  8729  fzrevral  9916  nfsum1  11157  nfsum  11158  fsum2dlemstep  11235  fisumcom2  11239  nfcprod1  11355  nfcprod  11356  bezoutlemmain  11722  cnmpt21  12499  setindis  13336  bdsetindis  13338  strcollnfALT  13355  isomninnlem  13400  iswomninnlem  13417  ismkvnnlem  13419