Theorem pm5.21im 644

Description: Two propositions are equivalent if they are both false. Closed form of 2false 649. Equivalent to a bi2 128-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.) (Revised by Mario Carneiro, 31-Jan-2015.)

Assertion

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

Proof of Theorem pm5.21im

Step	Hyp	Ref	Expression
1		pm5.21 643	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ 𝜓))
2	1	ex 113	1 ⊢ (¬ 𝜑 → (¬ 𝜓 → (𝜑 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  nbn2  645