![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfnf1 | Structured version Visualization version GIF version |
Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2166. (Revised by Wolf Lammen, 12-Oct-2021.) |
Ref | Expression |
---|---|
nfnf1 | ⊢ Ⅎ𝑥Ⅎ𝑥𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nf 1778 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
2 | nfe1 2139 | . . 3 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
3 | nfa1 2140 | . . 3 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
4 | 2, 3 | nfim 1891 | . 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑) |
5 | 1, 4 | nfxfr 1847 | 1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∃wex 1773 Ⅎwnf 1777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-10 2129 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ex 1774 df-nf 1778 |
This theorem is referenced by: nfsb4t 2492 nfnfc1 2894 nfabdwOLD 2916 sbcnestgfw 4420 sbcnestgf 4425 bj-sbf4 36448 wl-equsal1t 37140 wl-sbid2ft 37143 wl-sb8t 37150 wl-mo2tf 37169 wl-eutf 37171 wl-mo2t 37173 wl-mo3t 37174 wl-sb8eut 37176 wl-sb8eutv 37177 ichnfimlem 46940 ichnfim 46941 |
Copyright terms: Public domain | W3C validator |