![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfor | Structured version Visualization version GIF version |
Description: If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ∨ 𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nf.1 | ⊢ Ⅎ𝑥𝜑 |
nf.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
nfor | ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-or 848 | . 2 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
2 | nf.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | nfn 1855 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝜑 |
4 | nf.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
5 | 3, 4 | nfim 1894 | . 2 ⊢ Ⅎ𝑥(¬ 𝜑 → 𝜓) |
6 | 1, 5 | nfxfr 1850 | 1 ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 847 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 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 |
This theorem is referenced by: nf3or 1903 axi12 2704 axbnd 2705 nfun 4180 nfunOLD 4181 nfpr 4697 rabsnifsb 4727 disjxun 5146 fsuppmapnn0fiubex 14030 nfsum1 15723 nfsum 15724 nfcprod1 15941 nfcprod 15942 fdc1 37733 dvdsrabdioph 42798 mnringmulrcld 44224 disjinfi 45135 iundjiun 46416 |
Copyright terms: Public domain | W3C validator |