Theorem xornbidc 1265

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 1264	. . . 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 1252	. . 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 616  DECID wdc 733   ⊻ wxo 1251

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 532  ax-in2 533  ax-io 617

This theorem depends on definitions:  df-bi 110  df-dc 734  df-xor 1252

This theorem is referenced by:  xordc  1266  xordidc  1273