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Theorem nfrexdxy 2351
 Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexdya 2353 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
Hypotheses
Ref Expression
nfraldxy.2 yφ
nfraldxy.3 (φxA)
nfraldxy.4 (φ → Ⅎxψ)
Assertion
Ref Expression
nfrexdxy (φ → Ⅎxy A ψ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)   A(x,y)

Proof of Theorem nfrexdxy
StepHypRef Expression
1 df-rex 2306 . 2 (y A ψy(y A ψ))
2 nfraldxy.2 . . 3 yφ
3 nfcv 2175 . . . . . 6 xy
43a1i 9 . . . . 5 (φxy)
5 nfraldxy.3 . . . . 5 (φxA)
64, 5nfeld 2190 . . . 4 (φ → Ⅎx y A)
7 nfraldxy.4 . . . 4 (φ → Ⅎxψ)
86, 7nfand 1457 . . 3 (φ → Ⅎx(y A ψ))
92, 8nfexd 1641 . 2 (φ → Ⅎxy(y A ψ))
101, 9nfxfrd 1361 1 (φ → Ⅎxy A ψ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  Ⅎwnf 1346  ∃wex 1378   ∈ wcel 1390  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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306 This theorem is referenced by:  nfrexdya  2353  nfrexxy  2355  nfunid  3578  strcollnft  9444
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