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Mirrors > Home > MPE Home > Th. List > dvelimv | Structured version Visualization version GIF version |
Description: Similar to dvelim 2450 with first hypothesis replaced by a distinct variable condition. Usage of this theorem is discouraged because it depends on ax-13 2371. Check out dvelimhw 2342 for a version requiring fewer axioms. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 30-Apr-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dvelimv.1 | ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
dvelimv | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1914 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | dvelimv.1 | . 2 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | dvelim 2450 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀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 |
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: dveeq2ALT 2453 dveel1 2460 dveel2 2461 rgen2a 3343 |
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