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Mirrors > Home > ILE Home > Th. List > ax12 | GIF version |
Description: Rederive the original version of the axiom from ax-i12 1487. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Ref | Expression |
---|---|
ax12 | ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax12or 1488 | . . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | |
2 | 1 | ori 713 | . . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
3 | 2 | ord 714 | . 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
4 | ax-4 1490 | . 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))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 698 ∀wal 1333 = wceq 1335 |
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 605 ax-io 699 ax-i12 1487 ax-4 1490 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: (None) |
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