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Mirrors > Home > MPE Home > Th. List > Mathboxes > equidqe | Structured version Visualization version GIF version |
Description: equid 2019 with existential quantifier without using ax-c5 36021 or ax-5 1911. (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 36034 | . 2 ⊢ ¬ ∀𝑦 ¬ 𝑦 = 𝑥 | |
2 | ax7 2023 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑥 → 𝑥 = 𝑥)) | |
3 | 2 | pm2.43i 52 | . . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑥) |
4 | 3 | con3i 157 | . . 3 ⊢ (¬ 𝑥 = 𝑥 → ¬ 𝑦 = 𝑥) |
5 | 4 | alimi 1812 | . 2 ⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑦 = 𝑥) |
6 | 1, 5 | mto 199 | 1 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-c7 36023 ax-c10 36024 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 |
This theorem is referenced by: axc5sp1 36061 equidq 36062 |
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