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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 1787 | . . 3 ⊢ (∃𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ ∃𝑦 ¬ 𝜑) | |
2 | alex 1832 | . . . 4 ⊢ (∀𝑦𝜑 ↔ ¬ ∃𝑦 ¬ 𝜑) | |
3 | 2 | albii 1826 | . . 3 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥 ¬ ∃𝑦 ¬ 𝜑) |
4 | 1, 3 | xchbinxr 338 | . 2 ⊢ (∃𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑥∀𝑦𝜑) |
5 | 4 | bicomi 227 | 1 ⊢ (¬ ∀𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 |
This theorem depends on definitions: df-bi 210 df-ex 1787 |
This theorem is referenced by: 2exanali 1867 spc2gv 3504 spc2d 3506 hashfun 13890 pm11.52 41543 |
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