Theorem pm5.16

Description: Theorem *5.16 of WhiteheadRussell p. 124. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 17-Oct-2013)

	Ref	Expression
Assertion	pm5.16	⊢ ¬ ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜑 ↔ ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.18	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ ( 𝜑 ↔ ¬ 𝜓 ) )
2	1	biimpi	⊢ ( ( 𝜑 ↔ 𝜓 ) → ¬ ( 𝜑 ↔ ¬ 𝜓 ) )
3		imnan	⊢ ( ( ( 𝜑 ↔ 𝜓 ) → ¬ ( 𝜑 ↔ ¬ 𝜓 ) ) ↔ ¬ ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜑 ↔ ¬ 𝜓 ) ) )
4	2 3	mpbi	⊢ ¬ ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜑 ↔ ¬ 𝜓 ) )