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Theorem nfrexya 2341
Description: Not-free for restricted existential quantification where y and A are distinct. See nfrexxy 2339 for a version with x and y distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
Hypotheses
Ref Expression
nfralya.1 xA
nfralya.2 xφ
Assertion
Ref Expression
nfrexya xy A φ
Distinct variable group:   y,A
Allowed substitution hints:   φ(x,y)   A(x)

Proof of Theorem nfrexya
StepHypRef Expression
1 nftru 1335 . . 3 y
2 nfralya.1 . . . 4 xA
32a1i 9 . . 3 ( ⊤ → xA)
4 nfralya.2 . . . 4 xφ
54a1i 9 . . 3 ( ⊤ → Ⅎxφ)
61, 3, 5nfrexdya 2337 . 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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290
This theorem is referenced by:  nfiunya  3659
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