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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 2978. (Revised by Wolf Lammen, 9-Dec-2019.) |
Ref | Expression |
---|---|
dfral2 | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotb 303 | . . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑) | |
2 | 1 | ralbii 2963 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑) |
3 | ralnex 2975 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑) | |
4 | 2, 3 | bitri 263 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∀wral 2896 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-ral 2901 df-rex 2902 |
This theorem is referenced by: rexnal 2978 boxcutc 7837 infssuni 8140 ac6n 9190 indstr 11632 trfil3 21502 tglowdim2ln 25346 nmobndseqi 27018 stri 28500 hstri 28508 bnj1204 30334 nosepon 31066 poimirlem1 32580 |
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