Theorem nfralxy 2430

Description: Not-free for restricted universal quantification where x and y are distinct. See nfralya 2432 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 1410	. . 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 2426	. 2 |- ( T. -> F/ x A. y e. A ph )
7	6	mptru 1308	1 |- F/ x A. y e. A ph

Syntax hints:   T. wtru 1300   F/wnf 1404   F/_wnfc 2227   A.wral 2375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-17 1474  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380

This theorem is referenced by:  nfra2xy  2434  rspc2  2754  sbcralt  2937  sbcralg  2939  raaanlem  3415  nfint  3728  nfiinxy  3787  nfpo  4161  nfso  4162  nfse  4201  nffrfor  4208  nfwe  4215  ralxpf  4623  funimaexglem  5142  fun11iun  5322  dff13f  5603  nfiso  5639  mpoeq123  5762  nfofr  5920  fmpox  6028  nfrecs  6134  xpf1o  6667  ac6sfi  6721  lble  8563  fzrevral  9726  nfsum1  10964  nfsum  10965  fsum2dlemstep  11042  fisumcom2  11046  bezoutlemmain  11479  cnmpt21  12241  setindis  12750  bdsetindis  12752  isomninnlem  12809