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Mirrors > Home > MPE Home > Th. List > nfimd | Structured version Visualization version GIF version |
Description: If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 → 𝜒). Deduction form of nfim 1893. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1781 changed. (Revised by Wolf Lammen, 18-Sep-2021.) Eliminate curried form of nfimt 1892. (Revised by Wolf Lammen, 10-Jul-2022.) |
Ref | Expression |
---|---|
nfimd.1 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
nfimd.2 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
Ref | Expression |
---|---|
nfimd | ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.35 1874 | . . . 4 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (∀𝑥𝜓 → ∃𝑥𝜒)) | |
2 | 1 | biimpi 218 | . . 3 ⊢ (∃𝑥(𝜓 → 𝜒) → (∀𝑥𝜓 → ∃𝑥𝜒)) |
3 | nfimd.1 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | 3 | nfrd 1788 | . . . 4 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) |
5 | nfimd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
6 | 5 | nfrd 1788 | . . . 4 ⊢ (𝜑 → (∃𝑥𝜒 → ∀𝑥𝜒)) |
7 | 4, 6 | imim12d 81 | . . 3 ⊢ (𝜑 → ((∀𝑥𝜓 → ∃𝑥𝜒) → (∃𝑥𝜓 → ∀𝑥𝜒))) |
8 | 19.38 1835 | . . 3 ⊢ ((∃𝑥𝜓 → ∀𝑥𝜒) → ∀𝑥(𝜓 → 𝜒)) | |
9 | 2, 7, 8 | syl56 36 | . 2 ⊢ (𝜑 → (∃𝑥(𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒))) |
10 | 9 | nfd 1787 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∃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 |
This theorem depends on definitions: df-bi 209 df-ex 1777 df-nf 1781 |
This theorem is referenced by: nfimt 1892 nfand 1894 nfbid 1899 nfim1 2195 hbimd 2302 dvelimhw 2362 dvelimf 2466 nfmod2 2638 nfmodv 2639 nfraldw 3223 nfrald 3224 nfifd 4494 nfixpw 8479 nfixp 8480 axrepndlem1 10013 axrepndlem2 10014 axunndlem1 10016 axunnd 10017 axpowndlem2 10019 axpowndlem3 10020 axpowndlem4 10021 axregndlem2 10024 axregnd 10025 axinfndlem1 10026 axinfnd 10027 axacndlem4 10031 axacndlem5 10032 axacnd 10033 bj-dvelimdv 34175 wl-mo2df 34805 wl-mo2t 34810 riotasv2d 36092 nfintd 44775 |
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