Theorem ax12 1526

Description: Rederive the original version of the axiom from ax-i12 1521. (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

Step	Hyp	Ref	Expression
1		ax12or 1522	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2	1	ori 724	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
3	2	ord 725	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
4		ax-4 1524	. 2 ⊢ (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
5	3, 4	syl6 33	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 709  ∀wal 1362   = wceq 1364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710  ax-i12 1521  ax-4 1524

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)