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Theorem bj-axext3 33099
Description: Remove dependency on ax-13 2407 from axext3 2752. (Contributed by BJ, 12-Jul-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axext3 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem bj-axext3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ2 2158 . . . . 5 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
21bibi1d 332 . . . 4 (𝑤 = 𝑥 → ((𝑧𝑤𝑧𝑦) ↔ (𝑧𝑥𝑧𝑦)))
32albidv 2000 . . 3 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑤𝑧𝑦) ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))
4 equequ1 2109 . . 3 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
53, 4imbi12d 333 . 2 (𝑤 = 𝑥 → ((∀𝑧(𝑧𝑤𝑧𝑦) → 𝑤 = 𝑦) ↔ (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)))
6 ax-ext 2750 . 2 (∀𝑧(𝑧𝑤𝑧𝑦) → 𝑤 = 𝑦)
75, 6bj-chvarvv 33057 1 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628
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-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852
This theorem is referenced by:  bj-axext4  33100
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