Theorem xordc 1353

Description: Two ways to express "exclusive or" between decidable propositions. Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.)

Assertion

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

Proof of Theorem xordc

Step	Hyp	Ref	Expression
1		excxor 1339	. . . 4 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)))
2		ancom 264	. . . . 5 ⊢ ((¬ 𝜑 ∧ 𝜓) ↔ (𝜓 ∧ ¬ 𝜑))
3	2	orbi2i 734	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
4	1, 3	bitri 183	. . 3 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
5		xornbidc 1352	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))))
6	5	imp 123	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)))
7	4, 6	syl5rbbr 194	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))))
8	7	ex 114	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 680  DECID wdc 802   ⊻ wxo 1336

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681

This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-xor 1337

This theorem is referenced by:  dfbi3dc  1358  pm5.24dc  1359