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Mirrors > Home > ILE Home > Th. List > dcbi | GIF version |
Description: An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.) |
Ref | Expression |
---|---|
dcbi | ⊢ (DECID φ → (DECID ψ → DECID (φ ↔ ψ))) |
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))) |
Colors of variables: wff set class |
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 |
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