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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax13fromc9 | Structured version Visualization version GIF version |
Description: Derive ax-13 2374 from ax-c9 36900 and other older axioms.
This proof uses newer axioms ax-4 1816 and ax-6 1975, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 36894 and ax-c10 36896. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax13fromc9 | ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-c5 36893 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | |
2 | 1 | con3i 154 | . . 3 ⊢ (¬ 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦) |
3 | ax-c5 36893 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → 𝑥 = 𝑧) | |
4 | 3 | con3i 154 | . . 3 ⊢ (¬ 𝑥 = 𝑧 → ¬ ∀𝑥 𝑥 = 𝑧) |
5 | ax-c9 36900 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))) | |
6 | 2, 4, 5 | syl2im 40 | . 2 ⊢ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))) |
7 | ax13b 2039 | . 2 ⊢ ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))) | |
8 | 6, 7 | mpbir 230 | 1 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-c5 36893 ax-c9 36900 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1787 |
This theorem is referenced by: (None) |
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