Theorem xor3dc 1275

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 745	. . . . . 6 ⊢ (DECID ψ → DECID ¬ ψ)
2		dcbi 843	. . . . . 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 776	. . . . . . 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 773	. . . 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 741

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 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by:  pm5.15dc  1277  xor2dc  1278  nbbndc  1282