Theorem pm5.21 685

Description: Two propositions are equivalent if they are both false. Theorem *5.21 of [WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)

Assertion

Ref	Expression
pm5.21	⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ 𝜓))

Proof of Theorem pm5.21

Step	Hyp	Ref	Expression
1		simpl 108	. . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜑)
2	1	pm2.21d 609	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑 → 𝜓))
3		simpr 109	. . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜓)
4	3	pm2.21d 609	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜓 → 𝜑))
5	2, 4	impbid 128	1 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ 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-in2 605

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm5.21im  686