Theorem xornbidc 1386

Description: Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.)

Assertion

Ref	Expression
xornbidc	⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))))

Proof of Theorem xornbidc

Step	Hyp	Ref	Expression
1		df-xor 1371	. . 3 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
2		xor2dc 1385	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))))
3	2	imp 123	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))))
4	1, 3	bitr4id 198	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)))
5	4	ex 114	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  DECID wdc 829   ⊻ wxo 1370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-xor 1371

This theorem is referenced by:  xordc  1387  xordidc  1394