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Mirrors > Home > MPE Home > Th. List > Mathboxes > equidqe | Structured version Visualization version GIF version |
Description: equid 1926 with existential quantifier without using ax-c5 33186 or ax-5 1827. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
equidqe | ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6fromc10 33199 | . 2 ⊢ ¬ ∀𝑦 ¬ 𝑦 = 𝑥 | |
2 | ax7 1930 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑥 → 𝑥 = 𝑥)) | |
3 | 2 | pm2.43i 50 | . . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑥) |
4 | 3 | con3i 149 | . . 3 ⊢ (¬ 𝑥 = 𝑥 → ¬ 𝑦 = 𝑥) |
5 | 4 | alimi 1730 | . 2 ⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑦 = 𝑥) |
6 | 1, 5 | mto 187 | 1 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-c7 33188 ax-c10 33189 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: axc5sp1 33226 equidq 33227 |
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