Theorem 2nalexn 1755

Description: Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
2nalexn	⊢ (¬ ∀𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦 ¬ 𝜑)

Proof of Theorem 2nalexn

Step	Hyp	Ref	Expression
1		df-ex 1705	. . 3 ⊢ (∃𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ ∃𝑦 ¬ 𝜑)
2		alex 1753	. . . 4 ⊢ (∀𝑦𝜑 ↔ ¬ ∃𝑦 ¬ 𝜑)
3	2	albii 1747	. . 3 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥 ¬ ∃𝑦 ¬ 𝜑)
4	1, 3	xchbinxr 325	. 2 ⊢ (∃𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑥∀𝑦𝜑)
5	4	bicomi 214	1 ⊢ (¬ ∀𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  spc2gv  3296  hashfun  13224  spc2d  29313  pm11.52  38586  2exanali  38587