Theorem xor2dc 1281

Description: Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.)

Assertion

Ref	Expression
xor2dc	⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))))

Proof of Theorem xor2dc

Step	Hyp	Ref	Expression
1		xor3dc 1278	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))
2	1	imp 115	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
3		pm5.17dc 810	. . . 4 ⊢ (DECID 𝜓 → (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)))
4	3	adantl 262	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)))
5	2, 4	bitr4d 180	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))))
6	5	ex 108	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))))

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

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

This theorem is referenced by:  xornbidc  1282