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Theorem zfauscl 3957
Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3955, we invoke the Axiom of Extensionality (indirectly via vtocl 2673), 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(𝑥𝑦 ↔ (𝑥𝐴𝜑))
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 2151 . . . . . 6 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
32anbi1d 453 . . . . 5 (𝑧 = 𝐴 → ((𝑥𝑧𝜑) ↔ (𝑥𝐴𝜑)))
43bibi2d 230 . . . 4 (𝑧 = 𝐴 → ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝐴𝜑))))
54albidv 1752 . . 3 (𝑧 = 𝐴 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
65exbidv 1753 . 2 (𝑧 = 𝐴 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
7 ax-sep 3955 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
81, 6, 7vtocl 2673 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070  ax-sep 3955
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  inex1  3971  bj-d0clsepcl  11703
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