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Mirrors > Home > MPE Home > Th. List > nfeqf1 | Structured version Visualization version GIF version |
Description: An equation between setvar is free of any other setvar. (Contributed by Wolf Lammen, 10-Jun-2019.) |
Ref | Expression |
---|---|
nfeqf1 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeqf2 2297 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) | |
2 | equcom 1945 | . . 3 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧) | |
3 | 2 | nfbii 1778 | . 2 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 ↔ Ⅎ𝑥 𝑦 = 𝑧) |
4 | 1, 3 | sylib 208 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 Ⅎwnf 1708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 |
This theorem is referenced by: dveeq1 2300 sbal2 2461 nfeud2 2482 nfiotad 5854 wl-mo2df 33352 wl-eudf 33354 |
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