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Theorem bj-ax9 32083
Description: Proof of ax-9 1986 from ax-ext 2590 and df-cleq 2603 (and FOL) (with two extra dv conditions on 𝑥, 𝑧 and 𝑦, 𝑧). This shows that df-cleq 2603 is "too powerful". A possible definition is given by bj-df-cleq 32085. (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
StepHypRef Expression
1 dfcleq 2604 . . 3 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
21biimpi 205 . 2 (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
3 sp 2041 . . 3 (∀𝑧(𝑧𝑥𝑧𝑦) → (𝑧𝑥𝑧𝑦))
4 biimp 204 . . 3 ((𝑧𝑥𝑧𝑦) → (𝑧𝑥𝑧𝑦))
53, 4syl 17 . 2 (∀𝑧(𝑧𝑥𝑧𝑦) → (𝑧𝑥𝑧𝑦))
62, 5syl 17 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-ex 1696  df-cleq 2603
This theorem is referenced by: (None)
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