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Theorem nfrexya 2357
 Description: Not-free for restricted existential quantification where y and A are distinct. See nfrexxy 2355 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 1352 . . 3 y
2 nfralya.1 . . . 4 xA
32a1i 9 . . 3 ( ⊤ → xA)
4 nfralya.2 . . . 4 xφ
54a1i 9 . . 3 ( ⊤ → Ⅎxφ)
61, 3, 5nfrexdya 2353 . 2 ( ⊤ → Ⅎxy A φ)
76trud 1251 1 xy A φ
 Colors of variables: wff set class Syntax hints:   ⊤ wtru 1243  Ⅎwnf 1346  Ⅎwnfc 2162  ∃wrex 2301 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306 This theorem is referenced by:  nfiunya  3676  nffrec  5921
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