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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 2383 using ax-c15 38890. (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 1910 | . 2 ⊢ (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤) | |
| 2 | ax-5 1910 | . 2 ⊢ (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦) | |
| 3 | equequ2 2025 | . 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦)) | |
| 4 | 1, 2, 3 | dvelimf-o 38930 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-c5 38884 ax-c4 38885 ax-c7 38886 ax-c10 38887 ax-c11 38888 ax-c9 38891 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: ax12eq 38942 ax12el 38943 ax12inda 38949 ax12v2-o 38950 |
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