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Theorem zfauscl 4743
Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4741, we invoke the Axiom of Extensionality (indirectly via vtocl 3245), 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 4800 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 2687 . . . . . 6 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
32anbi1d 740 . . . . 5 (𝑧 = 𝐴 → ((𝑥𝑧𝜑) ↔ (𝑥𝐴𝜑)))
43bibi2d 332 . . . 4 (𝑧 = 𝐴 → ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝐴𝜑))))
54albidv 1846 . . 3 (𝑧 = 𝐴 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
65exbidv 1847 . 2 (𝑧 = 𝐴 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
7 ax-sep 4741 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
81, 6, 7vtocl 3245 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-12 2044  ax-ext 2601  ax-sep 4741
This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1483  df-ex 1702  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3188
This theorem is referenced by:  inex1  4759
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