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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. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by Wolf Lammen, 10-Jun-2019.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfeqf1 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeqf2 2376 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) | |
2 | equcom 2022 | . . 3 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧) | |
3 | 2 | nfbii 1855 | . 2 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 ↔ Ⅎ𝑥 𝑦 = 𝑧) |
4 | 1, 3 | sylib 217 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-13 2371 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 |
This theorem is referenced by: dveeq1 2379 sbal2 2533 nfmod2 2557 nfiotad 6458 wl-mo2df 36054 wl-eudf 36056 |
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