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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. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2177. (Revised by Wolf Lammen, 16-Dec-2022.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nfeqf2 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | exnal 1827 | . 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) | |
| 2 | hbe1 2143 | . . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → ∀𝑥∃𝑥 𝑧 = 𝑦) | |
| 3 | ax13lem2 2381 | . . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦)) | |
| 4 | ax13lem1 2379 | . . . . . 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 1864 | . . . 4 ⊢ (∃𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥∀𝑥 𝑧 = 𝑦)) |
| 7 | hbe1a 2144 | . . . 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 1790 | . 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) |
| 10 | 1, 9 | sylbir 235 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1538 ∃wex 1779 Ⅎwnf 1783 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: dveeq2 2383 nfeqf1 2384 sb4b 2480 sbal1 2533 copsexg 5496 axrepndlem1 10632 axpowndlem2 10638 axpowndlem3 10639 bj-dvelimdv 36852 bj-dvelimdv1 36853 wl-equsb3 37557 wl-sbcom2d-lem1 37560 wl-mo2df 37571 wl-eudf 37573 wl-euequf 37575 |
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