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Mirrors > Home > MPE Home > Th. List > dvelimh | Structured version Visualization version GIF version |
Description: Version of dvelim 2451 without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2372. Check out dvelimhw 2343 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 2142 | . . 3 ⊢ Ⅎ𝑥𝜑 |
3 | dvelimh.2 | . . . 4 ⊢ (𝜓 → ∀𝑧𝜓) | |
4 | 3 | nf5i 2142 | . . 3 ⊢ Ⅎ𝑧𝜓 |
5 | dvelimh.3 | . . 3 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 2, 4, 5 | dvelimf 2448 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓) |
7 | 6 | nf5rd 2189 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1537 |
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 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 |
This theorem is referenced by: dvelim 2451 dveeq1-o16 36950 dveel2ALT 36953 ax6e2nd 42178 ax6e2ndVD 42528 ax6e2ndALT 42550 |
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