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Theorem zfauscl 3868
Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3866, we invoke the Axiom of Extensionality (indirectly via vtocl 2602), which is needed for the justification of class variable notation. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
zfauscl.1 A V
Assertion
Ref Expression
zfauscl yx(x y ↔ (x A φ))
Distinct variable groups:   x,y,A   φ,y
Allowed substitution hint:   φ(x)

Proof of Theorem zfauscl
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 zfauscl.1 . 2 A V
2 eleq2 2098 . . . . . 6 (z = A → (x zx A))
32anbi1d 438 . . . . 5 (z = A → ((x z φ) ↔ (x A φ)))
43bibi2d 221 . . . 4 (z = A → ((x y ↔ (x z φ)) ↔ (x y ↔ (x A φ))))
54albidv 1702 . . 3 (z = A → (x(x y ↔ (x z φ)) ↔ x(x y ↔ (x A φ))))
65exbidv 1703 . 2 (z = A → (yx(x y ↔ (x z φ)) ↔ yx(x y ↔ (x A φ))))
7 ax-sep 3866 . 2 yx(x y ↔ (x z φ))
81, 6, 7vtocl 2602 1 yx(x y ↔ (x A φ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551
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-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019  ax-sep 3866
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  inex1  3882  bj-d0clsepcl  9314
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