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Theorem pm5.21 644
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) → (𝜑𝜓))

Proof of Theorem pm5.21
StepHypRef Expression
1 simpl 107 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜑)
21pm2.21d 582 . 2 ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑𝜓))
3 simpr 108 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜓)
43pm2.21d 582 . 2 ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜓𝜑))
52, 4impbid 127 1 ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  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 578
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  pm5.21im  645
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