Theorem pm11.52

Description: Theorem *11.52 in WhiteheadRussell p. 164. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm11.52	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ¬ ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		df-an	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ¬ ( 𝜑 → ¬ 𝜓 ) )
2	1	2exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ∃ 𝑦 ¬ ( 𝜑 → ¬ 𝜓 ) )
3		2nalexn	⊢ ( ¬ ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ¬ 𝜓 ) ↔ ∃ 𝑥 ∃ 𝑦 ¬ ( 𝜑 → ¬ 𝜓 ) )
4	2 3	bitr4i	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ¬ ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ¬ 𝜓 ) )