Theorem dcbi 843

Description: An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)

Assertion

Ref	Expression
dcbi	⊢ (DECID φ → (DECID ψ → DECID (φ ↔ ψ)))

Proof of Theorem dcbi

Step	Hyp	Ref	Expression
1		dcim 783	. . 3 ⊢ (DECID φ → (DECID ψ → DECID (φ → ψ)))
2		dcim 783	. . . 4 ⊢ (DECID ψ → (DECID φ → DECID (ψ → φ)))
3	2	com12 27	. . 3 ⊢ (DECID φ → (DECID ψ → DECID (ψ → φ)))
4		dcan 841	. . 3 ⊢ (DECID (φ → ψ) → (DECID (ψ → φ) → DECID ((φ → ψ) ∧ (ψ → φ))))
5	1, 3, 4	syl6c 60	. 2 ⊢ (DECID φ → (DECID ψ → DECID ((φ → ψ) ∧ (ψ → φ))))
6		dfbi2 368	. . 3 ⊢ ((φ ↔ ψ) ↔ ((φ → ψ) ∧ (ψ → φ)))
7	6	dcbii 746	. 2 ⊢ (DECID (φ ↔ ψ) ↔ DECID ((φ → ψ) ∧ (ψ → φ)))
8	5, 7	syl6ibr 151	1 ⊢ (DECID φ → (DECID ψ → DECID (φ ↔ ψ)))

Syntax hints:   → 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:  xor3dc  1275  pm5.15dc  1277  bilukdc  1284  xordidc  1287