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Theorem nfralxy 2377
Description: Not-free for restricted universal quantification where  x and  y are distinct. See nfralya 2379 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 1371 . . 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 2373 . 2  |-  ( T. 
->  F/ x A. y  e.  A  ph )
76trud 1268 1  |-  F/ x A. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:   T. wtru 1260   F/wnf 1365   F/_wnfc 2181   A.wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328
This theorem is referenced by:  nfra2xy  2381  rspc2  2683  sbcralt  2861  sbcralg  2863  raaanlem  3353  nfint  3652  nfiinxy  3711  nfpo  4065  nfso  4066  nfse  4105  nffrfor  4112  nfwe  4119  ralxpf  4509  funimaexglem  5009  fun11iun  5174  dff13f  5436  nfiso  5473  mpt2eq123  5591  fmpt2x  5853  nfrecs  5952  ac6sfi  6382  fzrevral  9068  setindis  10458  bdsetindis  10460
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