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Theorem zfauscl 4016
Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4014, we invoke the Axiom of Extensionality (indirectly via vtocl 2712), 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 2179 . . . . . 6 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
32anbi1d 458 . . . . 5 (𝑧 = 𝐴 → ((𝑥𝑧𝜑) ↔ (𝑥𝐴𝜑)))
43bibi2d 231 . . . 4 (𝑧 = 𝐴 → ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝐴𝜑))))
54albidv 1778 . . 3 (𝑧 = 𝐴 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
65exbidv 1779 . 2 (𝑧 = 𝐴 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
7 ax-sep 4014 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
81, 6, 7vtocl 2712 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wal 1312   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097  ax-sep 4014
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660
This theorem is referenced by:  inex1  4030  bj-d0clsepcl  12925
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