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Mirrors > Home > MPE Home > Th. List > ax13dgen2 | Structured version Visualization version GIF version |
Description: Degenerate instance of ax-13 2246 where bundled variables 𝑥 and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |
Ref | Expression |
---|---|
ax13dgen2 | ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equcomi 1944 | . 2 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
2 | pm2.21 120 | . 2 ⊢ (¬ 𝑥 = 𝑦 → (𝑥 = 𝑦 → ∀𝑥 𝑦 = 𝑥)) | |
3 | 1, 2 | syl5 34 | 1 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: (None) |
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