![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > xordc | GIF version |
Description: Two ways to express "exclusive or" between decidable propositions. Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
Ref | Expression |
---|---|
xordc | ⊢ (DECID φ → (DECID ψ → (¬ (φ ↔ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excxor 1268 | . . . 4 ⊢ ((φ ⊻ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (¬ φ ∧ ψ))) | |
2 | ancom 253 | . . . . 5 ⊢ ((¬ φ ∧ ψ) ↔ (ψ ∧ ¬ φ)) | |
3 | 2 | orbi2i 678 | . . . 4 ⊢ (((φ ∧ ¬ ψ) ∨ (¬ φ ∧ ψ)) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ))) |
4 | 1, 3 | bitri 173 | . . 3 ⊢ ((φ ⊻ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ))) |
5 | xornbidc 1279 | . . . 4 ⊢ (DECID φ → (DECID ψ → ((φ ⊻ ψ) ↔ ¬ (φ ↔ ψ)))) | |
6 | 5 | imp 115 | . . 3 ⊢ ((DECID φ ∧ DECID ψ) → ((φ ⊻ ψ) ↔ ¬ (φ ↔ ψ))) |
7 | 4, 6 | syl5rbbr 184 | . 2 ⊢ ((DECID φ ∧ DECID ψ) → (¬ (φ ↔ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ)))) |
8 | 7 | ex 108 | 1 ⊢ (DECID φ → (DECID ψ → (¬ (φ ↔ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ))))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 628 DECID wdc 741 ⊻ wxo 1265 |
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 df-xor 1266 |
This theorem is referenced by: dfbi3dc 1285 pm5.24dc 1286 |
Copyright terms: Public domain | W3C validator |