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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 1904. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1792 changed. (Revised by Wolf Lammen, 18-Sep-2021.) Eliminate curried form of nfimt 1903. (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 1885 | . . . 4 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (∀𝑥𝜓 → ∃𝑥𝜒)) | |
2 | 1 | biimpi 219 | . . 3 ⊢ (∃𝑥(𝜓 → 𝜒) → (∀𝑥𝜓 → ∃𝑥𝜒)) |
3 | nfimd.1 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | 3 | nfrd 1799 | . . . 4 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) |
5 | nfimd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
6 | 5 | nfrd 1799 | . . . 4 ⊢ (𝜑 → (∃𝑥𝜒 → ∀𝑥𝜒)) |
7 | 4, 6 | imim12d 81 | . . 3 ⊢ (𝜑 → ((∀𝑥𝜓 → ∃𝑥𝜒) → (∃𝑥𝜓 → ∀𝑥𝜒))) |
8 | 19.38 1846 | . . 3 ⊢ ((∃𝑥𝜓 → ∀𝑥𝜒) → ∀𝑥(𝜓 → 𝜒)) | |
9 | 2, 7, 8 | syl56 36 | . 2 ⊢ (𝜑 → (∃𝑥(𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒))) |
10 | 9 | nfd 1798 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 ∃wex 1787 Ⅎwnf 1791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 |
This theorem depends on definitions: df-bi 210 df-ex 1788 df-nf 1792 |
This theorem is referenced by: nfimt 1903 nfand 1905 nfbid 1910 nfim1 2198 hbimd 2300 dvelimhw 2347 dvelimf 2448 nfmod2 2558 nfmodv 2559 nfabdw 2928 nfraldw 3145 nfraldwOLD 3146 nfrald 3147 nfifd 4483 nfixpw 8618 nfixp 8619 axrepndlem1 10231 axrepndlem2 10232 axunndlem1 10234 axunnd 10235 axpowndlem2 10237 axpowndlem3 10238 axpowndlem4 10239 axregndlem2 10242 axregnd 10243 axinfndlem1 10244 axinfnd 10245 axacndlem4 10249 axacndlem5 10250 axacnd 10251 bj-dvelimdv 34798 wl-mo2df 35489 wl-mo2t 35494 riotasv2d 36735 nfintd 46079 |
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