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Mirrors > Home > MPE Home > Th. List > exnal | Structured version Visualization version GIF version |
Description: Existential quantification of negation is equivalent to negation of universal quantification. Dual of alnex 1784. See also the dual pair df-ex 1783 / alex 1829. Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
Ref | Expression |
---|---|
exnal | ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alex 1829 | . 2 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑) | |
2 | 1 | con2bii 358 | 1 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1540 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-ex 1783 |
This theorem is referenced by: alexn 1848 nfnbi 1858 exanali 1863 19.35 1881 19.30 1885 nfeqf2 2377 nabbib 3046 r2exlem 3144 spc3gv 3595 vn0 4339 notzfaus 5362 dtruALT2 5369 dvdemo1 5372 dtruALT 5387 eunex 5389 reusv2lem2 5398 dtru 5437 dtruOLD 5442 brprcneu 6882 brprcneuALT 6883 dffv2 6987 zfcndpow 10611 hashfun 14397 nmo 31730 bnj1304 33830 bnj1253 34028 axrepprim 34671 axunprim 34672 axregprim 34674 axinfprim 34675 axacprim 34676 dftr6 34721 brtxpsd 34866 elfuns 34887 dfrdg4 34923 relowlpssretop 36245 onsupmaxb 41988 clsk3nimkb 42791 expandexn 43048 vk15.4j 43289 vk15.4jVD 43675 alneu 45832 |
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