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Mirrors > Home > MPE Home > Th. List > nfald | Structured version Visualization version GIF version |
Description: Deduction form of nfal 2322. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 16-Oct-2021.) |
Ref | Expression |
---|---|
nfald.1 | ⊢ Ⅎ𝑦𝜑 |
nfald.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfald | ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.12 2326 | . . 3 ⊢ (∃𝑥∀𝑦𝜓 → ∀𝑦∃𝑥𝜓) | |
2 | nfald.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | nfald.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | 3 | nfrd 1788 | . . . 4 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) |
5 | 2, 4 | alimd 2210 | . . 3 ⊢ (𝜑 → (∀𝑦∃𝑥𝜓 → ∀𝑦∀𝑥𝜓)) |
6 | ax-11 2155 | . . 3 ⊢ (∀𝑦∀𝑥𝜓 → ∀𝑥∀𝑦𝜓) | |
7 | 1, 5, 6 | syl56 36 | . 2 ⊢ (𝜑 → (∃𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)) |
8 | 7 | nfd 1787 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 ∃wex 1776 Ⅎwnf 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-11 2155 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-or 848 df-ex 1777 df-nf 1781 |
This theorem is referenced by: nfexd 2328 dvelimhw 2346 nfald2 2448 nfmodv 2557 nfeqd 2914 nfabdw 2925 nfraldw 3307 nfiotadw 6519 nfixpw 8955 axrepndlem1 10630 axrepndlem2 10631 axunnd 10634 axpowndlem2 10636 axpowndlem4 10638 axregndlem2 10641 axinfndlem1 10643 axinfnd 10644 axacndlem4 10648 axacndlem5 10649 axacnd 10650 axsepg2 35075 axsepg2ALT 35076 axnulg 35085 bj-dvelimdv 36834 wl-mo2df 37551 wl-eudf 37553 wl-mo2t 37556 nfintd 48904 |
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