Theorem tsan3 33120

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		pm3.14 522	. . . 4 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓))
2	1	olcs 409	. . 3 ⊢ (¬ 𝜓 → ¬ (𝜑 ∧ 𝜓))
3	2	orri 390	. 2 ⊢ (𝜓 ∨ ¬ (𝜑 ∧ 𝜓))
4	3	a1i 11	1 ⊢ (𝜃 → (𝜓 ∨ ¬ (𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383

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-an 385

This theorem is referenced by:  tsna3  33123  ts3an3  33129  mpt2bi123f  33141  mptbi12f  33145