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Theorem nfrexya 2449
Description: Not-free for restricted existential quantification where  y and  A are distinct. See nfrexxy 2447 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
nfrexya  |-  F/ x E. y  e.  A  ph
Distinct variable group:    y, A
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem nfrexya
StepHypRef Expression
1 nftru 1425 . . 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, 5nfrexdya 2445 . 2  |-  ( T. 
->  F/ x E. y  e.  A  ph )
76mptru 1323 1  |-  F/ x E. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:   T. wtru 1315   F/wnf 1419   F/_wnfc 2243   E.wrex 2392
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397
This theorem is referenced by:  nfiunya  3809  nffrec  6259  nfsup  6845  caucvgsrlemgt1  7567  nfsum1  11076  zsupcllemstep  11545  bezout  11606
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