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

Proof of Theorem nfrexdya
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-rex 2306 . 2 (y A ψy(y A ψ))
2 sban 1826 . . . . . 6 ([z / y](y A ψ) ↔ ([z / y]y A [z / y]ψ))
3 clelsb3 2139 . . . . . . 7 ([z / y]y Az A)
43anbi1i 431 . . . . . 6 (([z / y]y A [z / y]ψ) ↔ (z A [z / y]ψ))
52, 4bitri 173 . . . . 5 ([z / y](y A ψ) ↔ (z A [z / y]ψ))
65exbii 1493 . . . 4 (z[z / y](y A ψ) ↔ z(z A [z / y]ψ))
7 nfv 1418 . . . . 5 z(y A ψ)
87sb8e 1734 . . . 4 (y(y A ψ) ↔ z[z / y](y A ψ))
9 df-rex 2306 . . . 4 (z A [z / y]ψz(z A [z / y]ψ))
106, 8, 93bitr4i 201 . . 3 (y(y A ψ) ↔ z A [z / y]ψ)
11 nfv 1418 . . . 4 zφ
12 nfraldya.3 . . . 4 (φxA)
13 nfraldya.2 . . . . 5 yφ
14 nfraldya.4 . . . . 5 (φ → Ⅎxψ)
1513, 14nfsbd 1848 . . . 4 (φ → Ⅎx[z / y]ψ)
1611, 12, 15nfrexdxy 2351 . . 3 (φ → Ⅎxz A [z / y]ψ)
1710, 16nfxfrd 1361 . 2 (φ → Ⅎxy(y A ψ))
181, 17nfxfrd 1361 1 (φ → Ⅎxy A ψ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  Ⅎwnf 1346  ∃wex 1378   ∈ wcel 1390  [wsb 1642  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-bnd 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-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306 This theorem is referenced by:  nfrexya  2357
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