![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dveeq2ALT | Structured version Visualization version GIF version |
Description: Alternate proof of dveeq2 2385, shorter but requiring ax-11 2158. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
dveeq2ALT | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equequ2 2033 | . 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦)) | |
2 | 1 | dvelimv 2463 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-11 2158 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |