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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 2363. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2163. (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 1821 | . 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) | |
2 | hbe1 2131 | . . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → ∀𝑥∃𝑥 𝑧 = 𝑦) | |
3 | ax13lem2 2367 | . . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦)) | |
4 | ax13lem1 2365 | . . . . . 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 1859 | . . . 4 ⊢ (∃𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥∀𝑥 𝑧 = 𝑦)) |
7 | hbe1a 2132 | . . . 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 1784 | . 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) |
10 | 1, 9 | sylbir 234 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1531 ∃wex 1773 Ⅎwnf 1777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-13 2363 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-nf 1778 |
This theorem is referenced by: dveeq2 2369 nfeqf1 2370 sb4b 2466 sbal1 2519 copsexg 5482 axrepndlem1 10584 axpowndlem2 10590 axpowndlem3 10591 bj-dvelimdv 36231 bj-dvelimdv1 36232 wl-equsb3 36925 wl-sbcom2d-lem1 36928 wl-mo2df 36939 wl-eudf 36941 wl-euequf 36943 |
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