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Theorem bj-axsep2 33046
Description: Remove dependency on ax-12 2087 and ax-13 2282 from axsep2 4815 while shortening its proof. Remark: the comment in zfauscl 4816 is misleading: the essential use of ax-ext 2631 is the one via eleq2 2719 and not the one via vtocl 3290, since the latter can be proved without ax-ext 2631 (see bj-vtocl 33034). (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axsep2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑧)

Proof of Theorem bj-axsep2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ2 2044 . . . . . 6 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
21anbi1d 741 . . . . 5 (𝑤 = 𝑧 → ((𝑥𝑤𝜑) ↔ (𝑥𝑧𝜑)))
32bibi2d 331 . . . 4 (𝑤 = 𝑧 → ((𝑥𝑦 ↔ (𝑥𝑤𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝑧𝜑))))
43albidv 1889 . . 3 (𝑤 = 𝑧 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑤𝜑)) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
54exbidv 1890 . 2 (𝑤 = 𝑧 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤𝜑)) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
6 ax-sep 4814 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤𝜑))
75, 6bj-chvarvv 32851 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-sep 4814
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by: (None)
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