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Theorem bj-ax9 32874
 Description: Proof of ax-9 1998 from Tarski's FOL=, sp 2052, df-cleq 2614 and ax-ext 2601 (with two extra dv conditions on 𝑥, 𝑧 and 𝑦, 𝑧). For a version without these dv conditions, see bj-ax9-2 32875. This shows that df-cleq 2614 is "too powerful". A possible definition is given by bj-df-cleq 32877. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ax9 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem bj-ax9
Dummy variables 𝑤 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-ext 2601 . . 3 (∀𝑧(𝑧𝑢𝑧𝑤) → 𝑢 = 𝑤)
21df-cleq 2614 . 2 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
3 biimp 205 . . 3 ((𝑧𝑥𝑧𝑦) → (𝑧𝑥𝑧𝑦))
43sps 2054 . 2 (∀𝑧(𝑧𝑥𝑧𝑦) → (𝑧𝑥𝑧𝑦))
52, 4sylbi 207 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-ex 1704  df-cleq 2614 This theorem is referenced by: (None)
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