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Mirrors > Home > MPE Home > Th. List > nfand | 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). (Contributed by Mario Carneiro, 7-Oct-2016.) |
Ref | Expression |
---|---|
nfand.1 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
nfand.2 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
Ref | Expression |
---|---|
nfand | ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-an 399 | . 2 ⊢ ((𝜓 ∧ 𝜒) ↔ ¬ (𝜓 → ¬ 𝜒)) | |
2 | nfand.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
3 | nfand.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
4 | 3 | nfnd 1858 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜒) |
5 | 2, 4 | nfimd 1895 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → ¬ 𝜒)) |
6 | 5 | nfnd 1858 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ (𝜓 → ¬ 𝜒)) |
7 | 1, 6 | nfxfrd 1854 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 |
This theorem is referenced by: nf3and 1899 nfan 1900 nfbid 1903 nfeud2 2676 nfeudw 2677 nfeld 2989 nfreud 3372 nfrmod 3373 nfreuw 3374 nfrmow 3375 nfrmo 3377 nfrabw 3385 nfrab 3386 nfifd 4495 nfdisjw 5043 nfdisj 5044 dfid3 5462 nfriotadw 7122 nfriotad 7125 axrepndlem1 10014 axrepndlem2 10015 axunndlem1 10017 axunnd 10018 axregndlem2 10025 axinfndlem1 10027 axinfnd 10028 axacndlem4 10032 axacndlem5 10033 axacnd 10034 cbvreud 34657 riotasv2d 36108 |
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