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Theorem bj-zfauscl 34130
Description: General version of zfauscl 5201.

Remark: the comment in zfauscl 5201 is misleading: the essential use of ax-ext 2797 is the one via eleq2 2905 and not the one via vtocl 3564, since the latter can be proved without ax-ext 2797 (see bj-vtoclg 34123).

(Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.)

Assertion
Ref Expression
bj-zfauscl (𝐴𝑉 → ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem bj-zfauscl
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2905 . . . . . . 7 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
21anbi1d 629 . . . . . 6 (𝑧 = 𝐴 → ((𝑥𝑧𝜑) ↔ (𝑥𝐴𝜑)))
32bibi2d 344 . . . . 5 (𝑧 = 𝐴 → ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝐴𝜑))))
43biimpd 230 . . . 4 (𝑧 = 𝐴 → ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) → (𝑥𝑦 ↔ (𝑥𝐴𝜑))))
54alimdv 1910 . . 3 (𝑧 = 𝐴 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) → ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
65eximdv 1911 . 2 (𝑧 = 𝐴 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) → ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
7 ax-sep 5199 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
86, 7bj-vtoclg 34123 1 (𝐴𝑉 → ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1528   = wceq 1530  wex 1773  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2797  ax-sep 5199
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-cleq 2818  df-clel 2897
This theorem is referenced by: (None)
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