Theorem xordc 1280

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 1268	. . . 4 ⊢ ((φ ⊻ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (¬ φ ∧ ψ)))
2		ancom 253	. . . . 5 ⊢ ((¬ φ ∧ ψ) ↔ (ψ ∧ ¬ φ))
3	2	orbi2i 678	. . . 4 ⊢ (((φ ∧ ¬ ψ) ∨ (¬ φ ∧ ψ)) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ)))
4	1, 3	bitri 173	. . 3 ⊢ ((φ ⊻ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ)))
5		xornbidc 1279	. . . 4 ⊢ (DECID φ → (DECID ψ → ((φ ⊻ ψ) ↔ ¬ (φ ↔ ψ))))
6	5	imp 115	. . 3 ⊢ ((DECID φ ∧ DECID ψ) → ((φ ⊻ ψ) ↔ ¬ (φ ↔ ψ)))
7	4, 6	syl5rbbr 184	. 2 ⊢ ((DECID φ ∧ DECID ψ) → (¬ (φ ↔ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ))))
8	7	ex 108	1 ⊢ (DECID φ → (DECID ψ → (¬ (φ ↔ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628  DECID wdc 741   ⊻ wxo 1265

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  df-xor 1266

This theorem is referenced by:  dfbi3dc  1285  pm5.24dc  1286