Theorem tsxo1 38145

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

Assertion

Ref	Expression
tsxo1	⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊻ 𝜓)))

Proof of Theorem tsxo1

Step	Hyp	Ref	Expression
1		tsbi1 38141	. 2 ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓)))
2		xnor 1512	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))
3	2	orbi2i 912	. 2 ⊢ (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊻ 𝜓)))
4	1, 3	sylib 218	1 ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊻ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847   ⊻ wxo 1510

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 207  df-an 396  df-or 848  df-xor 1511

This theorem is referenced by: (None)