Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dveeq2-o | Structured version Visualization version GIF version |
Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2387 using ax-c15 35905. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dveeq2-o | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1902 | . 2 ⊢ (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤) | |
2 | ax-5 1902 | . 2 ⊢ (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦) | |
3 | equequ2 2024 | . 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦)) | |
4 | 1, 2, 3 | dvelimf-o 35945 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 ax-c5 35899 ax-c4 35900 ax-c7 35901 ax-c10 35902 ax-c11 35903 ax-c9 35906 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 |
This theorem is referenced by: ax12eq 35957 ax12el 35958 ax12inda 35964 ax12v2-o 35965 |
Copyright terms: Public domain | W3C validator |