Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dvdemo2 | Structured version Visualization version GIF version |
Description: Demonstration of a theorem (scheme) that requires (meta)variables 𝑥 and 𝑧 to be distinct, but no others. It bundles the theorem schemes ∃𝑥(𝑥 = 𝑥 → 𝑧 ∈ 𝑥) and ∃𝑥(𝑥 = 𝑦 → 𝑦 ∈ 𝑥). Compare dvdemo1 4902. (Contributed by NM, 1-Dec-2006.) |
Ref | Expression |
---|---|
dvdemo2 | ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | el 4847 | . 2 ⊢ ∃𝑥 𝑧 ∈ 𝑥 | |
2 | ax-1 6 | . 2 ⊢ (𝑧 ∈ 𝑥 → (𝑥 = 𝑦 → 𝑧 ∈ 𝑥)) | |
3 | 1, 2 | eximii 1764 | 1 ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-11 2034 ax-12 2047 ax-13 2246 ax-pow 4843 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |