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Mirrors > Home > MPE Home > Th. List > dveel1 | Structured version Visualization version GIF version |
Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Ref | Expression |
---|---|
dveel1 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → ∀𝑥 𝑦 ∈ 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elequ1 2151 | . 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) | |
2 | 1 | dvelimv 2487 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → ∀𝑥 𝑦 ∈ 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-tru 1633 df-ex 1852 df-nf 1857 |
This theorem is referenced by: distel 32039 |
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