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Theorem bdzfauscl 7263
Description: Closed form of the version of zfauscl 3851 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 2083 . . . . . 6 (z = A → (x zx A))
21anbi1d 441 . . . . 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 1687 . . 3 (z = A → (x(x y ↔ (x z φ)) ↔ x(x y ↔ (x A φ))))
54exbidv 1688 . 2 (z = A → (yx(x y ↔ (x z φ)) ↔ yx(x y ↔ (x A φ))))
6 bdzfauscl.bd . . 3 BOUNDED φ
76bdsep2 7259 . 2 yx(x y ↔ (x z φ))
85, 7vtoclg 2590 1 (A 𝑉yx(x y ↔ (x A φ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1226   = wceq 1228  wex 1362   wcel 1374  BOUNDED wbd 7186
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  ax-bdsep 7258
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537
This theorem is referenced by:  bdinex1  7269
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