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Theorem nfralya 2530
Description: Not-free for restricted universal quantification where  y and  A are distinct. See nfralxy 2528 for a version with  x and  y distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
Hypotheses
Ref Expression
nfralya.1  |-  F/_ x A
nfralya.2  |-  F/ x ph
Assertion
Ref Expression
nfralya  |-  F/ x A. y  e.  A  ph
Distinct variable group:    y, A
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem nfralya
StepHypRef Expression
1 nftru 1477 . . 3  |-  F/ y T.
2 nfralya.1 . . . 4  |-  F/_ x A
32a1i 9 . . 3  |-  ( T. 
->  F/_ x A )
4 nfralya.2 . . . 4  |-  F/ x ph
54a1i 9 . . 3  |-  ( T. 
->  F/ x ph )
61, 3, 5nfraldya 2525 . 2  |-  ( T. 
->  F/ x A. y  e.  A  ph )
76mptru 1373 1  |-  F/ x A. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:   T. wtru 1365   F/wnf 1471   F/_wnfc 2319   A.wral 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473
This theorem is referenced by:  nfiinya  3930  nfsup  7021  caucvgsrlemgt1  7824  axpre-suploclemres  7930  supinfneg  9625  infsupneg  9626  ctiunctlemudc  12488  trirec0  15251
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