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| Description: Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) | 
| Ref | Expression | 
|---|---|
| 2nalexn | ⊢ (¬ ∀𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦 ¬ 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ex 1780 | . . 3 ⊢ (∃𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ ∃𝑦 ¬ 𝜑) | |
| 2 | alex 1826 | . . . 4 ⊢ (∀𝑦𝜑 ↔ ¬ ∃𝑦 ¬ 𝜑) | |
| 3 | 2 | albii 1819 | . . 3 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥 ¬ ∃𝑦 ¬ 𝜑) | 
| 4 | 1, 3 | xchbinxr 335 | . 2 ⊢ (∃𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑥∀𝑦𝜑) | 
| 5 | 4 | bicomi 224 | 1 ⊢ (¬ ∀𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦 ¬ 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∀wal 1538 ∃wex 1779 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 | 
| This theorem is referenced by: 2exanali 1860 spc2gv 3600 spc2d 3602 hashfun 14476 ralopabb 43424 pm11.52 44406 | 
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