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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 2377 using ax-c15 37401. (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 1914 | . 2 ⊢ (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤) | |
2 | ax-5 1914 | . 2 ⊢ (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦) | |
3 | equequ2 2030 | . 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦)) | |
4 | 1, 2, 3 | dvelimf-o 37441 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2371 ax-c5 37395 ax-c4 37396 ax-c7 37397 ax-c10 37398 ax-c11 37399 ax-c9 37402 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 |
This theorem is referenced by: ax12eq 37453 ax12el 37454 ax12inda 37460 ax12v2-o 37461 |
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