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Theorem bj-ax9-2 33016
Description: Proof of ax-9 2039 from Tarski's FOL=, ax-8 2032 (specifically, ax8v1 2034 and ax8v2 2035) , df-cleq 2644 and ax-ext 2631. For a version not using ax-8 2032, see bj-ax9 33015. This shows that df-cleq 2644 is "too powerful". A possible definition is given by bj-df-cleq 33018. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ax9-2 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Proof of Theorem bj-ax9-2
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-ext 2631 . . . . 5 (∀𝑢(𝑢𝑣𝑢𝑤) → 𝑣 = 𝑤)
21df-cleq 2644 . . . 4 (𝑥 = 𝑦 ↔ ∀𝑢(𝑢𝑥𝑢𝑦))
32biimpi 206 . . 3 (𝑥 = 𝑦 → ∀𝑢(𝑢𝑥𝑢𝑦))
4 biimp 205 . . 3 ((𝑢𝑥𝑢𝑦) → (𝑢𝑥𝑢𝑦))
53, 4sylg 1790 . 2 (𝑥 = 𝑦 → ∀𝑢(𝑢𝑥𝑢𝑦))
6 ax8v2 2035 . . . . 5 (𝑧 = 𝑢 → (𝑧𝑥𝑢𝑥))
76equcoms 1993 . . . 4 (𝑢 = 𝑧 → (𝑧𝑥𝑢𝑥))
8 ax8v1 2034 . . . 4 (𝑢 = 𝑧 → (𝑢𝑦𝑧𝑦))
97, 8imim12d 81 . . 3 (𝑢 = 𝑧 → ((𝑢𝑥𝑢𝑦) → (𝑧𝑥𝑧𝑦)))
109spimvw 1973 . 2 (∀𝑢(𝑢𝑥𝑢𝑦) → (𝑧𝑥𝑧𝑦))
115, 10syl 17 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521
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  ax-8 2032  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-cleq 2644
This theorem is referenced by: (None)
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