Theorem xornbidc 1282

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		xor2dc 1281	. . . 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 1267	. . 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))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742   ⊻ wxo 1266

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 630

This theorem depends on definitions:  df-bi 110  df-dc 743  df-xor 1267

This theorem is referenced by:  xordc  1283  xordidc  1290