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Theorem nfralxy 2471
Description: Not-free for restricted universal quantification where  x and  y are distinct. See nfralya 2473 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
StepHypRef Expression
1 nftru 1442 . . 3  |-  F/ y T.
2 nfralxy.1 . . . 4  |-  F/_ x A
32a1i 9 . . 3  |-  ( T. 
->  F/_ x A )
4 nfralxy.2 . . . 4  |-  F/ x ph
54a1i 9 . . 3  |-  ( T. 
->  F/ x ph )
61, 3, 5nfraldxy 2467 . 2  |-  ( T. 
->  F/ x A. y  e.  A  ph )
76mptru 1340 1  |-  F/ x A. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:   T. wtru 1332   F/wnf 1436   F/_wnfc 2268   A.wral 2416
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421
This theorem is referenced by:  nfra2xy  2475  rspc2  2800  sbcralt  2985  sbcralg  2987  raaanlem  3468  nfint  3781  nfiinxy  3840  nfpo  4223  nfso  4224  nfse  4263  nffrfor  4270  nfwe  4277  ralxpf  4685  funimaexglem  5206  fun11iun  5388  dff13f  5671  nfiso  5707  mpoeq123  5830  nfofr  5988  fmpox  6098  nfrecs  6204  xpf1o  6738  ac6sfi  6792  ismkvnex  7029  lble  8717  fzrevral  9897  nfsum1  11137  nfsum  11138  fsum2dlemstep  11215  fisumcom2  11219  nfcprod1  11335  nfcprod  11336  bezoutlemmain  11697  cnmpt21  12474  setindis  13249  bdsetindis  13251  isomninnlem  13311
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