Theorem tsan3 35987

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

Assertion

Ref	Expression
tsan3	⊢ (𝜃 → (𝜓 ∨ ¬ (𝜑 ∧ 𝜓)))

Proof of Theorem tsan3

Step	Hyp	Ref	Expression
1		pm3.14 996	. . . 4 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓))
2	1	olcs 876	. . 3 ⊢ (¬ 𝜓 → ¬ (𝜑 ∧ 𝜓))
3	2	orri 862	. 2 ⊢ (𝜓 ∨ ¬ (𝜑 ∧ 𝜓))
4	3	a1i 11	1 ⊢ (𝜃 → (𝜓 ∨ ¬ (𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 847

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 210  df-an 400  df-or 848

This theorem is referenced by:  tsna3  35990  ts3an3  35996  mpobi123f  36006  mptbi12f  36010