![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 397 | . 2 ⊢ ((𝜓 ∧ 𝜒) ↔ ¬ (𝜓 → ¬ 𝜒)) | |
2 | nfand.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
3 | nfand.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
4 | 3 | nfnd 1861 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜒) |
5 | 2, 4 | nfimd 1897 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → ¬ 𝜒)) |
6 | 5 | nfnd 1861 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ (𝜓 → ¬ 𝜒)) |
7 | 1, 6 | nfxfrd 1856 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒)) |
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 |
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: nf3and 1901 nfan 1902 nfbid 1905 nfeud2 2584 nfeudw 2585 nfeld 2914 nfreuwOLD 3422 nfrmowOLD 3423 nfrmod 3428 nfreud 3429 nfrmo 3430 nfrabwOLD 3469 nfrab 3472 nfifd 4557 nfdisjw 5125 nfdisj 5126 nfopabd 5216 dfid3 5577 nfriotadw 7372 nfriotad 7376 axrepndlem1 10586 axrepndlem2 10587 axunndlem1 10589 axunnd 10590 axregndlem2 10597 axinfndlem1 10599 axinfnd 10600 axacndlem4 10604 axacndlem5 10605 axacnd 10606 bj-gabima 35815 cbvreud 36249 riotasv2d 37822 |
Copyright terms: Public domain | W3C validator |