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Theorem wl-ax8clv2 33511
Description: Axiom ax-wl-8cl 33507 carries over to our new definition df-wl-clelv2 33510. (Contributed by Wolf Lammen, 27-Nov-2021.)
Assertion
Ref Expression
wl-ax8clv2 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))

Proof of Theorem wl-ax8clv2
Dummy variables 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 equvinv 2003 . 2 (𝑥 = 𝑦 ↔ ∃𝑢(𝑢 = 𝑥𝑢 = 𝑦))
2 df-wl-clelv2 33510 . . . . 5 (𝑥𝐴 ↔ ∀𝑡(𝑡 = 𝑥𝑡𝐴))
3 equtrr 1995 . . . . . . 7 (𝑢 = 𝑥 → (𝑡 = 𝑢𝑡 = 𝑥))
43imim1d 82 . . . . . 6 (𝑢 = 𝑥 → ((𝑡 = 𝑥𝑡𝐴) → (𝑡 = 𝑢𝑡𝐴)))
54alimdv 1885 . . . . 5 (𝑢 = 𝑥 → (∀𝑡(𝑡 = 𝑥𝑡𝐴) → ∀𝑡(𝑡 = 𝑢𝑡𝐴)))
62, 5syl5bi 232 . . . 4 (𝑢 = 𝑥 → (𝑥𝐴 → ∀𝑡(𝑡 = 𝑢𝑡𝐴)))
7 equeuclr 1996 . . . . . . 7 (𝑢 = 𝑦 → (𝑡 = 𝑦𝑡 = 𝑢))
87imim1d 82 . . . . . 6 (𝑢 = 𝑦 → ((𝑡 = 𝑢𝑡𝐴) → (𝑡 = 𝑦𝑡𝐴)))
98alimdv 1885 . . . . 5 (𝑢 = 𝑦 → (∀𝑡(𝑡 = 𝑢𝑡𝐴) → ∀𝑡(𝑡 = 𝑦𝑡𝐴)))
10 df-wl-clelv2 33510 . . . . 5 (𝑦𝐴 ↔ ∀𝑡(𝑡 = 𝑦𝑡𝐴))
119, 10syl6ibr 242 . . . 4 (𝑢 = 𝑦 → (∀𝑡(𝑡 = 𝑢𝑡𝐴) → 𝑦𝐴))
126, 11sylan9 690 . . 3 ((𝑢 = 𝑥𝑢 = 𝑦) → (𝑥𝐴𝑦𝐴))
1312exlimiv 1898 . 2 (∃𝑢(𝑢 = 𝑥𝑢 = 𝑦) → (𝑥𝐴𝑦𝐴))
141, 13sylbi 207 1 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521  wex 1744  wcel-wl 33503  wcel2-wl 33505
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
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-wl-clelv2 33510
This theorem is referenced by: (None)
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