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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 2390. (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 2147 | . . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → ∀𝑥∃𝑥 𝑧 = 𝑦) | |
3 | ax13lem2 2394 | . . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦)) | |
4 | ax13lem1 2392 | . . . . . 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 1865 | . . . 4 ⊢ (∃𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥∀𝑥 𝑧 = 𝑦)) |
7 | hbe1a 2148 | . . . 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 1791 | . 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) |
10 | 1, 9 | sylbir 237 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1535 ∃wex 1780 Ⅎwnf 1784 |
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-10 2145 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 |
This theorem is referenced by: dveeq2 2396 nfeqf1 2397 sb4b 2499 sb4bOLD 2500 sbal1 2572 copsexg 5382 axrepndlem1 10014 axpowndlem2 10020 axpowndlem3 10021 bj-dvelimdv 34175 bj-dvelimdv1 34176 wl-equsb3 34807 wl-sbcom2d-lem1 34810 wl-mo2df 34821 wl-eudf 34823 wl-euequf 34825 |
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