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Theorem 2nalexn 1745
Description: Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
2nalexn (¬ ∀𝑥𝑦𝜑 ↔ ∃𝑥𝑦 ¬ 𝜑)

Proof of Theorem 2nalexn
StepHypRef Expression
1 df-ex 1696 . . 3 (∃𝑥𝑦 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ ∃𝑦 ¬ 𝜑)
2 alex 1743 . . . 4 (∀𝑦𝜑 ↔ ¬ ∃𝑦 ¬ 𝜑)
32albii 1737 . . 3 (∀𝑥𝑦𝜑 ↔ ∀𝑥 ¬ ∃𝑦 ¬ 𝜑)
41, 3xchbinxr 324 . 2 (∃𝑥𝑦 ¬ 𝜑 ↔ ¬ ∀𝑥𝑦𝜑)
54bicomi 213 1 (¬ ∀𝑥𝑦𝜑 ↔ ∃𝑥𝑦 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  wal 1473  wex 1695
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-ex 1696
This theorem is referenced by:  spc2gv  3269  hashfun  13084  spc2d  28697  pm11.52  37608  2exanali  37609
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