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Mirrors > Home > MPE Home > Th. List > dvdemo1 | Structured version Visualization version GIF version |
Description: Demonstration of a theorem (scheme) that requires (meta)variables 𝑥 and 𝑦 to be distinct, but no others. It bundles the theorem schemes ∃𝑥(𝑥 = 𝑦 → 𝑥 ∈ 𝑥) and ∃𝑥(𝑥 = 𝑦 → 𝑦 ∈ 𝑥). Compare dvdemo2 5086. ("Bundles" is a term introduced by Raph Levien.) (Contributed by NM, 1-Dec-2006.) |
Ref | Expression |
---|---|
dvdemo1 | ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dtru 5082 | . . 3 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 | |
2 | exnal 1870 | . . 3 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) | |
3 | 1, 2 | mpbir 223 | . 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦 |
4 | pm2.21 121 | . 2 ⊢ (¬ 𝑥 = 𝑦 → (𝑥 = 𝑦 → 𝑧 ∈ 𝑥)) | |
5 | 3, 4 | eximii 1880 | 1 ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1599 ∃wex 1823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-12 2162 ax-13 2333 ax-nul 5025 ax-pow 5077 |
This theorem depends on definitions: df-bi 199 df-an 387 df-tru 1605 df-ex 1824 df-nf 1828 |
This theorem is referenced by: (None) |
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