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Theorem zfauscl 3847
Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3845, we invoke the Axiom of Extensionality (indirectly via vtocl 2581), 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 2079 . . . . . 6 (z = A → (x zx A))
32anbi1d 441 . . . . 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 1683 . . 3 (z = A → (x(x y ↔ (x z φ)) ↔ x(x y ↔ (x A φ))))
65exbidv 1684 . 2 (z = A → (yx(x y ↔ (x z φ)) ↔ yx(x y ↔ (x A φ))))
7 ax-sep 3845 . 2 yx(x y ↔ (x z φ))
81, 6, 7vtocl 2581 1 yx(x y ↔ (x A φ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wal 1224   = wceq 1226  wex 1358   wcel 1370  Vcvv 2531
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 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-ext 2000  ax-sep 3845
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-v 2533
This theorem is referenced by:  inex1  3861  bj-d0clsepcl  7287
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