Theorem tsan2 38149

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		pm3.14 998	. . . 4 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓))
2	1	orcs 876	. . 3 ⊢ (¬ 𝜑 → ¬ (𝜑 ∧ 𝜓))
3	2	orri 863	. 2 ⊢ (𝜑 ∨ ¬ (𝜑 ∧ 𝜓))
4	3	a1i 11	1 ⊢ (𝜃 → (𝜑 ∨ ¬ (𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848

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 849

This theorem is referenced by:  tsna2  38152  ts3an2  38158  mpobi123f  38169  mptbi12f  38173