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Theorem bj-axsep2 32113
 Description: Remove dependency on ax-8 1979, ax-10 2006, ax-12 2034, ax-13 2234, ax-ext 2590, df-cleq 2603 and df-clel 2606 from axsep2 4710 while shortening its proof (note that axsep2 4710 does require ax-8 1979 and ax-9 1986 since it requires df-clel 2606 and df-cleq 2603--- see bj-df-clel 32081 and bj-df-cleq 32085). Remark: the comment in zfauscl 4711 is misleading: the essential use of ax-ext 2590 is the one via eleq2 2677 and not the one via vtocl 3232, since the latter can be proved without ax-ext 2590 (see bj-vtocl 32101). (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 1991 . . . . . 6 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
21anbi1d 737 . . . . 5 (𝑤 = 𝑧 → ((𝑥𝑤𝜑) ↔ (𝑥𝑧𝜑)))
32bibi2d 331 . . . 4 (𝑤 = 𝑧 → ((𝑥𝑦 ↔ (𝑥𝑤𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝑧𝜑))))
43albidv 1836 . . 3 (𝑤 = 𝑧 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑤𝜑)) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
54exbidv 1837 . 2 (𝑤 = 𝑧 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤𝜑)) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
6 ax-sep 4709 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤𝜑))
75, 6bj-chvarvv 31913 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
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