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Mirrors > Home > MPE Home > Th. List > dvelimh | Structured version Visualization version GIF version |
Description: Version of dvelim 2449 without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2370. Check out dvelimhw 2340 for a version requiring fewer axioms. (Contributed by NM, 1-Oct-2002.) (Proof shortened by Wolf Lammen, 11-May-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dvelimh.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
dvelimh.2 | ⊢ (𝜓 → ∀𝑧𝜓) |
dvelimh.3 | ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
dvelimh | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvelimh.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | 1 | nf5i 2141 | . . 3 ⊢ Ⅎ𝑥𝜑 |
3 | dvelimh.2 | . . . 4 ⊢ (𝜓 → ∀𝑧𝜓) | |
4 | 3 | nf5i 2141 | . . 3 ⊢ Ⅎ𝑧𝜓 |
5 | dvelimh.3 | . . 3 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 2, 4, 5 | dvelimf 2446 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓) |
7 | 6 | nf5rd 2188 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-10 2136 ax-11 2153 ax-12 2170 ax-13 2370 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 |
This theorem is referenced by: dvelim 2449 dveeq1-o16 38270 dveel2ALT 38273 ax6e2nd 43782 ax6e2ndVD 44132 ax6e2ndALT 44154 |
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