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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 2377. (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 2382 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) | |
| 2 | equcom 2017 | . . 3 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧) | |
| 3 | 2 | nfbii 1852 | . 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 1538 Ⅎwnf 1783 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: dveeq1 2385 sbal2 2534 nfmod2 2558 nfiotad 6519 wl-mo2df 37571 wl-eudf 37573 |
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