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Theorem nfrexxy 2339
Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2341 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
Hypotheses
Ref Expression
nfralxy.1 xA
nfralxy.2 xφ
Assertion
Ref Expression
nfrexxy xy A φ
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   A(x,y)

Proof of Theorem nfrexxy
StepHypRef Expression
1 nftru 1335 . . 3 y
2 nfralxy.1 . . . 4 xA
32a1i 9 . . 3 ( ⊤ → xA)
4 nfralxy.2 . . . 4 xφ
54a1i 9 . . 3 ( ⊤ → Ⅎxφ)
61, 3, 5nfrexdxy 2335 . 2 ( ⊤ → Ⅎxy A φ)
76trud 1237 1 xy A φ
Colors of variables: wff set class
Syntax hints:  wtru 1229  wnf 1329  wnfc 2147  wrex 2285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290
This theorem is referenced by:  r19.12  2400  sbcrext  2812  sbcrexg  2814  nfuni  3560  nfiunxy  3657  rexxpf  4410  abrexex2g  5670  abrexex2  5674  nfrecs  5844  bj-findis  7197  strcollnfALT  7204
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