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Theorem ax12 1505
Description: Rederive the original version of the axiom from ax-i12 1500. (Contributed by Mario Carneiro, 3-Feb-2015.)
Assertion
Ref Expression
ax12 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Proof of Theorem ax12
StepHypRef Expression
1 ax12or 1501 . . . 4 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
21ori 718 . . 3 (¬ ∀𝑧 𝑧 = 𝑥 → (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
32ord 719 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
4 ax-4 1503 . 2 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
53, 4syl6 33 1 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 703  wal 1346   = wceq 1348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 610  ax-io 704  ax-i12 1500  ax-4 1503
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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