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Mirrors > Home > MPE Home > Th. List > df-ex | Structured version Visualization version GIF version |
Description: Define existential quantification. ∃𝑥𝜑 means "there exists at least one set 𝑥 such that 𝜑 is true". Dual of alex 1822. See also the dual pair alnex 1777 / exnal 1823. Definition of [Margaris] p. 49. (Contributed by NM, 10-Jan-1993.) |
Ref | Expression |
---|---|
df-ex | ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | 1, 2 | wex 1775 | . 2 wff ∃𝑥𝜑 |
4 | 1 | wn 3 | . . . 4 wff ¬ 𝜑 |
5 | 4, 2 | wal 1534 | . . 3 wff ∀𝑥 ¬ 𝜑 |
6 | 5 | wn 3 | . 2 wff ¬ ∀𝑥 ¬ 𝜑 |
7 | 3, 6 | wb 206 | 1 wff (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) |
Colors of variables: wff setvar class |
This definition is referenced by: alnex 1777 eximal 1778 2nalexn 1824 2exnaln 1825 19.43OLD 1880 speimfw 1960 speimfwALT 1961 spimfw 1962 ax6ev 1966 cbvexvw 2033 hbe1w 2045 exexw 2048 hbe1 2140 hbe1a 2141 sbex 2279 nfex 2322 nfexd 2327 drex1v 2371 ax6 2386 drex1 2443 nfexd2 2448 eujustALT 2569 spcimegf 3550 spcegf 3591 spcimedv 3594 rexab 3702 neq0f 4353 neq0 4357 n0el 4369 abn0 4390 ax6vsep 5308 axnulALT 5309 exexneq 5444 axpownd 10638 gchi 10661 ballotlem2 34469 cbvex1v 35066 axextprim 35680 axrepprim 35681 axunprim 35682 axpowprim 35683 axinfprim 35685 axacprim 35686 distel 35784 bj-axtd 36576 bj-eximALT 36623 bj-modald 36655 bj-modalbe 36670 bj-hbext 36692 bj-cbvexdv 36782 bj-nfexd 37120 gneispace 44123 pm10.252 44356 hbexgVD 44903 elsetrecslem 48929 |
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