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Mirrors > Home > MPE Home > Th. List > dfral2 | Structured version Visualization version GIF version |
Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 3100. (Revised by Wolf Lammen, 9-Dec-2019.) |
Ref | Expression |
---|---|
dfral2 | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotb 315 | . . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑) | |
2 | 1 | ralbii 3093 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑) |
3 | ralnex 3072 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑) | |
4 | 2, 3 | bitri 275 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wral 3061 ∃wrex 3070 |
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-an 398 df-ex 1783 df-ral 3062 df-rex 3071 |
This theorem is referenced by: rexnal 3100 imaeqsalv 7310 boxcutc 8882 infssuni 9290 ac6n 10426 indstr 12846 trfil3 23255 nosepon 27029 noinfbnd1lem4 27090 tglowdim2ln 27635 nmobndseqi 29763 stri 31241 hstri 31249 reprinfz1 33292 bnj1204 33681 fvineqsneq 35929 poimirlem1 36125 n0elqs 36833 |
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