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Mirrors > Home > ILE Home > Th. List > equidqe | GIF version |
Description: equid 1629 with some quantification and negation without using ax-4 1440 or ax-17 1459. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) |
Ref | Expression |
---|---|
equidqe | ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-9 1464 | . 2 ⊢ ¬ ∀𝑦 ¬ 𝑦 = 𝑥 | |
2 | ax-8 1435 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑥 → 𝑥 = 𝑥)) | |
3 | 2 | pm2.43i 48 | . . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑥) |
4 | 3 | con3i 594 | . . 3 ⊢ (¬ 𝑥 = 𝑥 → ¬ 𝑦 = 𝑥) |
5 | 4 | alimi 1384 | . 2 ⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑦 = 𝑥) |
6 | 1, 5 | mto 620 | 1 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∀wal 1282 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-5 1376 ax-gen 1378 ax-ie2 1423 ax-8 1435 ax-i9 1463 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 |
This theorem is referenced by: ax4sp1 1466 |
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