Theorem pm5.16 860

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

Assertion

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

Proof of Theorem pm5.16

Step	Hyp	Ref	Expression
1		pm5.18 345	. . 3 ⊢ ((φ ↔ ψ) ↔ ¬ (φ ↔ ¬ ψ))
2	1	biimpi 186	. 2 ⊢ ((φ ↔ ψ) → ¬ (φ ↔ ¬ ψ))
3		imnan 411	. 2 ⊢ (((φ ↔ ψ) → ¬ (φ ↔ ¬ ψ)) ↔ ¬ ((φ ↔ ψ) ∧ (φ ↔ ¬ ψ)))
4	2, 3	mpbi 199	1 ⊢ ¬ ((φ ↔ ψ) ∧ (φ ↔ ¬ ψ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360

This theorem is used by: (None)