![]() |
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 2373 using ax-c15 38361. (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 1906 | . 2 ⊢ (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤) | |
2 | ax-5 1906 | . 2 ⊢ (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦) | |
3 | equequ2 2022 | . 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦)) | |
4 | 1, 2, 3 | dvelimf-o 38401 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-11 2147 ax-12 2167 ax-13 2367 ax-c5 38355 ax-c4 38356 ax-c7 38357 ax-c10 38358 ax-c11 38359 ax-c9 38362 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 |
This theorem is referenced by: ax12eq 38413 ax12el 38414 ax12inda 38420 ax12v2-o 38421 |
Copyright terms: Public domain | W3C validator |