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Mirrors > Home > MPE Home > Th. List > 2nalexn | Structured version Visualization version GIF version |
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 1783 | . . 3 ⊢ (∃𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ ∃𝑦 ¬ 𝜑) | |
2 | alex 1828 | . . . 4 ⊢ (∀𝑦𝜑 ↔ ¬ ∃𝑦 ¬ 𝜑) | |
3 | 2 | albii 1822 | . . 3 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥 ¬ ∃𝑦 ¬ 𝜑) |
4 | 1, 3 | xchbinxr 335 | . 2 ⊢ (∃𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑥∀𝑦𝜑) |
5 | 4 | bicomi 223 | 1 ⊢ (¬ ∀𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1537 ∃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: 2exanali 1863 spc2gv 3539 spc2d 3541 hashfun 14152 pm11.52 42005 |
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