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Theorem bj-axsep2 33246
 Description: Remove dependency on ax-12 2202 and ax-13 2407 from axsep2 4913 while shortening its proof. (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 2158 . . . . . 6 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
21anbi1d 607 . . . . 5 (𝑤 = 𝑧 → ((𝑥𝑤𝜑) ↔ (𝑥𝑧𝜑)))
32bibi2d 331 . . . 4 (𝑤 = 𝑧 → ((𝑥𝑦 ↔ (𝑥𝑤𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝑧𝜑))))
43albidv 2000 . . 3 (𝑤 = 𝑧 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑤𝜑)) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
54exbidv 2001 . 2 (𝑤 = 𝑧 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤𝜑)) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
6 ax-sep 4912 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤𝜑))
75, 6bj-chvarvv 33057 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 382  ∀wal 1628  ∃wex 1851 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-sep 4912 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852 This theorem is referenced by: (None)
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