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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 2375. (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 2380 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) | |
2 | equcom 2015 | . . 3 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧) | |
3 | 2 | nfbii 1849 | . 2 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 ↔ Ⅎ𝑥 𝑦 = 𝑧) |
4 | 1, 3 | sylib 218 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1535 Ⅎwnf 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-13 2375 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 |
This theorem is referenced by: dveeq1 2383 sbal2 2532 nfmod2 2556 nfiotad 6521 wl-mo2df 37551 wl-eudf 37553 |
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