Theorem pm4.63dc 779

Description: Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)

Assertion

Ref	Expression
pm4.63dc	⊢ (DECID φ → (DECID ψ → (¬ (φ → ¬ ψ) ↔ (φ ∧ ψ))))

Proof of Theorem pm4.63dc

Step	Hyp	Ref	Expression
1		dfandc 777	. . . 4 ⊢ (DECID φ → (DECID ψ → ((φ ∧ ψ) ↔ ¬ (φ → ¬ ψ))))
2	1	imp 115	. . 3 ⊢ ((DECID φ ∧ DECID ψ) → ((φ ∧ ψ) ↔ ¬ (φ → ¬ ψ)))
3	2	bicomd 129	. 2 ⊢ ((DECID φ ∧ DECID ψ) → (¬ (φ → ¬ ψ) ↔ (φ ∧ ψ)))
4	3	ex 108	1 ⊢ (DECID φ → (DECID ψ → (¬ (φ → ¬ ψ) ↔ (φ ∧ ψ))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  DECID wdc 741

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by:  pm4.67dc  780