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Mirrors > Home > MPE Home > Th. List > nfan1 | Structured version Visualization version GIF version |
Description: A closed form of nfan 1902. (Contributed by Mario Carneiro, 3-Oct-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.) (Proof shortened by Wolf Lammen, 7-Jul-2022.) |
Ref | Expression |
---|---|
nfim1.1 | ⊢ Ⅎ𝑥𝜑 |
nfim1.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfan1 | ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-an 397 | . 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓)) | |
2 | nfim1.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | nfim1.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | 3 | nfnd 1861 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓) |
5 | 2, 4 | nfim1 2192 | . . 3 ⊢ Ⅎ𝑥(𝜑 → ¬ 𝜓) |
6 | 5 | nfn 1860 | . 2 ⊢ Ⅎ𝑥 ¬ (𝜑 → ¬ 𝜓) |
7 | 1, 6 | nfxfr 1855 | 1 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 Ⅎwnf 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-nf 1786 |
This theorem is referenced by: sb4b 2474 ralcom2 3373 sbcralt 3866 sbcrext 3867 csbiebt 3923 riota5f 7393 axrepndlem1 10586 axrepndlem2 10587 axunnd 10590 axpowndlem2 10592 axpowndlem3 10593 axpowndlem4 10594 axregndlem2 10597 axinfndlem1 10599 axinfnd 10600 axacndlem4 10604 axacndlem5 10605 axacnd 10606 fproddivf 15930 wl-sbcom2d-lem1 36419 wl-mo2df 36430 wl-eudf 36432 wl-mo3t 36436 wl-ax11-lem4 36445 wl-ax11-lem6 36447 |
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