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Theorem ax9ALT 2754
 Description: Proof of ax-9 2122 from Tarski's FOL and dfcleq 2752. For a version not using ax-8 2114 either, see eleq2w2 2755. This shows that dfcleq 2752 is too powerful to be used as a definition instead of df-cleq 2751. Note that ax-ext 2730 is also a direct consequence of dfcleq 2752 (as an instance of its forward implication). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax9ALT (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Proof of Theorem ax9ALT
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2752 . . . 4 (𝑥 = 𝑦 ↔ ∀𝑡(𝑡𝑥𝑡𝑦))
21biimpi 219 . . 3 (𝑥 = 𝑦 → ∀𝑡(𝑡𝑥𝑡𝑦))
3 biimp 218 . . 3 ((𝑡𝑥𝑡𝑦) → (𝑡𝑥𝑡𝑦))
42, 3sylg 1825 . 2 (𝑥 = 𝑦 → ∀𝑡(𝑡𝑥𝑡𝑦))
5 ax8 2118 . . . . 5 (𝑧 = 𝑡 → (𝑧𝑥𝑡𝑥))
65equcoms 2028 . . . 4 (𝑡 = 𝑧 → (𝑧𝑥𝑡𝑥))
7 ax8 2118 . . . 4 (𝑡 = 𝑧 → (𝑡𝑦𝑧𝑦))
86, 7imim12d 81 . . 3 (𝑡 = 𝑧 → ((𝑡𝑥𝑡𝑦) → (𝑧𝑥𝑧𝑦)))
98spimvw 2003 . 2 (∀𝑡(𝑡𝑥𝑡𝑦) → (𝑧𝑥𝑧𝑦))
104, 9syl 17 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1537 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-cleq 2751 This theorem is referenced by: (None)
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