Theorem tsxo4

Description: A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018)

	Ref	Expression
Assertion	tsxo4	⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ⊻ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		tsbi4	⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) ) )
2		df-xor	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ¬ ( 𝜑 ↔ 𝜓 ) )
3	2	bicomi	⊢ ( ¬ ( 𝜑 ↔ 𝜓 ) ↔ ( 𝜑 ⊻ 𝜓 ) )
4	3	orbi2i	⊢ ( ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) ) ↔ ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ⊻ 𝜓 ) ) )
5	1 4	sylib	⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ⊻ 𝜓 ) ) )