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Mirrors > Home > MPE Home > Th. List > Mathboxes > dveeq1-o | Structured version Visualization version GIF version |
Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2288 using ax-c11 . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dveeq1-o | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1827 | . 2 ⊢ (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧) | |
2 | ax-5 1827 | . 2 ⊢ (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧) | |
3 | equequ1 1939 | . 2 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑦 = 𝑧)) | |
4 | 1, 2, 3 | dvelimf-o 33232 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-c5 33186 ax-c4 33187 ax-c7 33188 ax-c10 33189 ax-c11 33190 ax-c9 33193 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 |
This theorem is referenced by: ax12inda2ALT 33249 |
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