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Mirrors > Home > MPE Home > Th. List > nfeqf2 | Structured version Visualization version GIF version |
Description: An equation between setvar is free of any other setvar. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2167. (Revised by Wolf Lammen, 16-Dec-2022.) |
Ref | Expression |
---|---|
nfeqf2 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exnal 1818 | . 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) | |
2 | hbe1 2138 | . . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → ∀𝑥∃𝑥 𝑧 = 𝑦) | |
3 | ax13lem2 2387 | . . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦)) | |
4 | ax13lem1 2385 | . . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | |
5 | 3, 4 | syldc 48 | . . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → (¬ 𝑥 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
6 | 2, 5 | eximdh 1856 | . . . 4 ⊢ (∃𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥∀𝑥 𝑧 = 𝑦)) |
7 | hbe1a 2139 | . . . 4 ⊢ (∃𝑥∀𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) | |
8 | 6, 7 | syl6com 37 | . . 3 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
9 | 8 | nfd 1782 | . 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) |
10 | 1, 9 | sylbir 236 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1526 ∃wex 1771 Ⅎwnf 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-13 2383 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 |
This theorem is referenced by: dveeq2 2389 nfeqf1 2390 sb4b 2495 sb4bOLD 2496 sbal1 2568 copsexg 5374 axrepndlem1 10003 axpowndlem2 10009 axpowndlem3 10010 bj-dvelimdv 34073 bj-dvelimdv1 34074 wl-equsb3 34674 wl-sbcom2d-lem1 34677 wl-mo2df 34688 wl-eudf 34690 wl-euequf 34692 |
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