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Mirrors > Home > ILE Home > Th. List > xornbidc | GIF version |
Description: Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.) |
Ref | Expression |
---|---|
xornbidc | ⊢ (DECID φ → (DECID ψ → ((φ ⊻ ψ) ↔ ¬ (φ ↔ ψ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xor2dc 1278 | . . . 4 ⊢ (DECID φ → (DECID ψ → (¬ (φ ↔ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ))))) | |
2 | 1 | imp 115 | . . 3 ⊢ ((DECID φ ∧ DECID ψ) → (¬ (φ ↔ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ)))) |
3 | df-xor 1266 | . . 3 ⊢ ((φ ⊻ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ))) | |
4 | 2, 3 | syl6rbbr 188 | . 2 ⊢ ((DECID φ ∧ DECID ψ) → ((φ ⊻ ψ) ↔ ¬ (φ ↔ ψ))) |
5 | 4 | ex 108 | 1 ⊢ (DECID φ → (DECID ψ → ((φ ⊻ ψ) ↔ ¬ (φ ↔ ψ)))) |
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: xordc 1280 xordidc 1287 |
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