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Theorem zfauscl 5188
Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 5186, we invoke the Axiom of Extensionality (indirectly via vtocl 3544), which is needed for the justification of class variable notation.

If we omit the requirement that 𝑦 not occur in 𝜑, we can derive a contradiction, as notzfaus 5245 shows. (Contributed by NM, 21-Jun-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 2904 . . . . . 6 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
32anbi1d 632 . . . . 5 (𝑧 = 𝐴 → ((𝑥𝑧𝜑) ↔ (𝑥𝐴𝜑)))
43bibi2d 346 . . . 4 (𝑧 = 𝐴 → ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝐴𝜑))))
54albidv 1922 . . 3 (𝑧 = 𝐴 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
65exbidv 1923 . 2 (𝑧 = 𝐴 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
7 ax-sep 5186 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
81, 6, 7vtocl 3544 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2115  Vcvv 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796  ax-sep 5186
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817  df-clel 2896
This theorem is referenced by:  inex1  5204
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