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Theorem bj-ax9 33015
 Description: Proof of ax-9 2039 from Tarski's FOL=, sp 2091, df-cleq 2644 and ax-ext 2631 (with two extra dv conditions on 𝑥, 𝑧 and 𝑦, 𝑧). For a version without these dv conditions, see bj-ax9-2 33016. 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 (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 2631 . . 3 (∀𝑧(𝑧𝑢𝑧𝑤) → 𝑢 = 𝑤)
21df-cleq 2644 . 2 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
3 biimp 205 . . 3 ((𝑧𝑥𝑧𝑦) → (𝑧𝑥𝑧𝑦))
43sps 2093 . 2 (∀𝑧(𝑧𝑥𝑧𝑦) → (𝑧𝑥𝑧𝑦))
52, 4sylbi 207 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-12 2087  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-ex 1745  df-cleq 2644 This theorem is referenced by: (None)
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