Theorem xor3dc 1365

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 827	. . . . . 6 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
2		dcbi 920	. . . . . 6 ⊢ (DECID 𝜑 → (DECID ¬ 𝜓 → DECID (𝜑 ↔ ¬ 𝜓)))
3	1, 2	syl5 32	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ↔ ¬ 𝜓)))
4	3	imp 123	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ↔ ¬ 𝜓))
5		pm5.18dc 868	. . . . . . 7 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
6	5	imp 123	. . . . . 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 865	. . . 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 140	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
11	10	ex 114	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 819

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by:  pm5.15dc  1367  xor2dc  1368  nbbndc  1372