Theorem xor3dc 1278

Description: Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.)

Assertion

Ref	Expression
xor3dc	⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))

Proof of Theorem xor3dc

Step	Hyp	Ref	Expression
1		dcn 746	. . . . . 6 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
2		dcbi 844	. . . . . 6 ⊢ (DECID 𝜑 → (DECID ¬ 𝜓 → DECID (𝜑 ↔ ¬ 𝜓)))
3	1, 2	syl5 28	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ↔ ¬ 𝜓)))
4	3	imp 115	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ↔ ¬ 𝜓))
5		pm5.18dc 777	. . . . . . 7 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
6	5	imp 115	. . . . . 6 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
7	6	a1d 22	. . . . 5 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (DECID (𝜑 ↔ ¬ 𝜓) → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
8	7	con2biddc 774	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (DECID (𝜑 ↔ ¬ 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))))
9	4, 8	mpd 13	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)))
10	9	bicomd 129	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
11	10	ex 108	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  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:  pm5.15dc  1280  xor2dc  1281  nbbndc  1285