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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-dveeq12 | Structured version Visualization version GIF version |
Description: The current form of ax-13 2234 has a particular disadvantage: The condition ¬ 𝑥 = 𝑦 is less versatile than the general form ¬ ∀𝑥𝑥 = 𝑦. You need ax-10 2006 to arrive at the more general form presented here. You need 19.8a 2039 (or ax-12 2034) to restore 𝑦 = 𝑧 from ∃𝑥𝑦 = 𝑧 again. (Contributed by Wolf Lammen, 9-Jun-2021.) |
Ref | Expression |
---|---|
wl-dveeq12 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exnal 1744 | . 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) | |
2 | hbe1 2008 | . . 3 ⊢ (∃𝑥 𝑧 = 𝑦 → ∀𝑥∃𝑥 𝑧 = 𝑦) | |
3 | ax13lem2 2284 | . . . . . . 7 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦)) | |
4 | ax13lem1 2236 | . . . . . . 7 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | |
5 | 3, 4 | syld 46 | . . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
6 | 5 | com12 32 | . . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → (¬ 𝑥 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
7 | 6 | aleximi 1749 | . . . 4 ⊢ (∀𝑥∃𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥∀𝑥 𝑧 = 𝑦)) |
8 | 7 | com12 32 | . . 3 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥∃𝑥 𝑧 = 𝑦 → ∃𝑥∀𝑥 𝑧 = 𝑦)) |
9 | hbe1a 2009 | . . 3 ⊢ (∃𝑥∀𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) | |
10 | 2, 8, 9 | syl56 35 | . 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
11 | 1, 10 | sylbir 224 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 ∃wex 1695 |
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-10 2006 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: (None) |
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