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Theorem bj-axext4 31958
Description: Remove dependency on ax-13 2234 from axext4 2594. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axext4 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem bj-axext4
StepHypRef Expression
1 bj-elequ2g 31853 . 2 (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
2 bj-axext3 31957 . 2 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
31, 2impbii 198 1 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wal 1473
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-ext 2590
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696
This theorem is referenced by: (None)
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