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Theorem bdzfauscl 9345
 Description: Closed form of the version of zfauscl 3868 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
Hypothesis
Ref Expression
bdzfauscl.bd BOUNDED φ
Assertion
Ref Expression
bdzfauscl (A 𝑉yx(x y ↔ (x A φ)))
Distinct variable groups:   x,y,A   φ,y
Allowed substitution hints:   φ(x)   𝑉(x,y)

Proof of Theorem bdzfauscl
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 eleq2 2098 . . . . . 6 (z = A → (x zx A))
21anbi1d 438 . . . . 5 (z = A → ((x z φ) ↔ (x A φ)))
32bibi2d 221 . . . 4 (z = A → ((x y ↔ (x z φ)) ↔ (x y ↔ (x A φ))))
43albidv 1702 . . 3 (z = A → (x(x y ↔ (x z φ)) ↔ x(x y ↔ (x A φ))))
54exbidv 1703 . 2 (z = A → (yx(x y ↔ (x z φ)) ↔ yx(x y ↔ (x A φ))))
6 bdzfauscl.bd . . 3 BOUNDED φ
76bdsep1 9340 . 2 yx(x y ↔ (x z φ))
85, 7vtoclg 2607 1 (A 𝑉yx(x y ↔ (x A φ)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  BOUNDED wbd 9267 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  ax-bdsep 9339 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553 This theorem is referenced by:  bdinex1  9354
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