Theorem tsxo3 33116

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

Assertion

Ref	Expression
tsxo3	⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ⊻ 𝜓)))

Proof of Theorem tsxo3

Step	Hyp	Ref	Expression
1		tsbi3 33112	. 2 ⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))
2		df-xor 1457	. . . 4 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
3	2	bicomi 213	. . 3 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ⊻ 𝜓))
4	3	orbi2i 540	. 2 ⊢ (((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)) ↔ ((𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ⊻ 𝜓)))
5	1, 4	sylib 207	1 ⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ⊻ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ⊻ wxo 1456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-or 384  df-xor 1457

This theorem is referenced by: (None)