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Theorem bj-zfauscl 33226
Description: General version of zfauscl 4970, removing dependency on ax-12 2213 and df-clab 2789 (and df-tru 1641, df-sb 2060, df-v 3389).

Remark: the comment in zfauscl 4970 is misleading: the essential use of ax-ext 2781 is the one via eleq2 2870 and not the one via vtocl 3448, since the latter can be proved without ax-ext 2781 (see bj-vtoclg 33216).

(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 2870 . . . . . . 7 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
21anbi1d 617 . . . . . 6 (𝑧 = 𝐴 → ((𝑥𝑧𝜑) ↔ (𝑥𝐴𝜑)))
32bibi2d 333 . . . . 5 (𝑧 = 𝐴 → ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝐴𝜑))))
43biimpd 220 . . . 4 (𝑧 = 𝐴 → ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) → (𝑥𝑦 ↔ (𝑥𝐴𝜑))))
54alimdv 2007 . . 3 (𝑧 = 𝐴 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) → ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
65eximdv 2008 . 2 (𝑧 = 𝐴 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) → ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
7 ax-sep 4968 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
86, 7bj-vtoclg 33216 1 (𝐴𝑉 → ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  wex 1859  wcel 2155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-ext 2781  ax-sep 4968
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-cleq 2795  df-clel 2798
This theorem is referenced by: (None)
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