Theorem tsan2 38824

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

Assertion

Ref	Expression
tsan2	⊢ (𝜃 → (𝜑 ∨ ¬ (𝜑 ∧ 𝜓)))

Proof of Theorem tsan2

Step	Hyp	Ref	Expression
1		pm3.14 1011	. . . 4 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓))
2	1	orcs 889	. . 3 ⊢ (¬ 𝜑 → ¬ (𝜑 ∧ 𝜓))
3	2	orri 876	. 2 ⊢ (𝜑 ∨ ¬ (𝜑 ∧ 𝜓))
4	3	a1i 11	1 ⊢ (𝜃 → (𝜑 ∨ ¬ (𝜑 ∧ 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862

This theorem is used by:  tsna2  38827  ts3an2  38833  mpobi123f  38844  mptbi12f  38848