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| Description: Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1779. See also the dual pair alnex 1780 / exnal 1826. Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.) | 
| Ref | Expression | 
|---|---|
| alex | ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | notnotb 315 | . . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑) | |
| 2 | 1 | albii 1818 | . 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑) | 
| 3 | alnex 1780 | . 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑) | |
| 4 | 2, 3 | bitri 275 | 1 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∀wal 1537 ∃wex 1778 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 | 
| This theorem depends on definitions: df-bi 207 df-ex 1779 | 
| This theorem is referenced by: exnal 1826 2nalexn 1827 alimex 1830 emptyal 1907 nfa1 2150 sp 2182 exists2 2661 pm10.253 44386 vk15.4j 44553 vk15.4jVD 44939 | 
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