Theorem pm5.16 818

Description: Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)

Assertion

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

Proof of Theorem pm5.16

Step	Hyp	Ref	Expression
1		pm5.19 696	. 2 ⊢ ¬ (𝜓 ↔ ¬ 𝜓)
2		simpl 108	. . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓)) → (𝜑 ↔ 𝜓))
3		simpr 109	. . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓)) → (𝜑 ↔ ¬ 𝜓))
4	2, 3	bitr3d 189	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓)) → (𝜓 ↔ ¬ 𝜓))
5	1, 4	mto 652	1 ⊢ ¬ ((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)