![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dveeq1-o16 | Structured version Visualization version GIF version |
Description: Version of dveeq1 2378 using ax-c16 37567 instead of ax-5 1913. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1913. (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dveeq1-o16 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax5eq 37607 | . 2 ⊢ (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧) | |
2 | ax5eq 37607 | . 2 ⊢ (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧) | |
3 | equequ1 2028 | . 2 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑦 = 𝑧)) | |
4 | 1, 2, 3 | dvelimh 2448 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1539 |
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 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2370 ax-c9 37565 ax-c16 37567 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |