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Theorem bj-ax9-2 33220
 Description: Proof of ax-9 2154 from Tarski's FOL=, ax-8 2147 (specifically, ax8v1 2149 and ax8v2 2150) , df-cleq 2764 and ax-ext 2751. For a version not using ax-8 2147, see bj-ax9 33219. This shows that df-cleq 2764 is "too powerful". A possible definition is given by bj-df-cleq 33222. (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 2751 . . . . 5 (∀𝑢(𝑢𝑣𝑢𝑤) → 𝑣 = 𝑤)
21df-cleq 2764 . . . 4 (𝑥 = 𝑦 ↔ ∀𝑢(𝑢𝑥𝑢𝑦))
32biimpi 206 . . 3 (𝑥 = 𝑦 → ∀𝑢(𝑢𝑥𝑢𝑦))
4 biimp 205 . . 3 ((𝑢𝑥𝑢𝑦) → (𝑢𝑥𝑢𝑦))
53, 4sylg 1898 . 2 (𝑥 = 𝑦 → ∀𝑢(𝑢𝑥𝑢𝑦))
6 ax8v2 2150 . . . . 5 (𝑧 = 𝑢 → (𝑧𝑥𝑢𝑥))
76equcoms 2105 . . . 4 (𝑢 = 𝑧 → (𝑧𝑥𝑢𝑥))
8 ax8v1 2149 . . . 4 (𝑢 = 𝑧 → (𝑢𝑦𝑧𝑦))
97, 8imim12d 81 . . 3 (𝑢 = 𝑧 → ((𝑢𝑥𝑢𝑦) → (𝑧𝑥𝑧𝑦)))
109spimvw 2085 . 2 (∀𝑢(𝑢𝑥𝑢𝑦) → (𝑧𝑥𝑧𝑦))
115, 10syl 17 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1629 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-cleq 2764 This theorem is referenced by: (None)
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