Theorem tsbi1 37491

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

Assertion

Ref	Expression
tsbi1	⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓)))

Proof of Theorem tsbi1

Step	Hyp	Ref	Expression
1		pm5.1 821	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓))
2	1	olcd 871	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓)))
3		pm3.13 991	. . . 4 ⊢ (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
4	3	orcd 870	. . 3 ⊢ (¬ (𝜑 ∧ 𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓)))
5	2, 4	pm2.61i 182	. 2 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓))
6	5	a1i 11	1 ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844

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 206  df-an 396  df-or 845

This theorem is referenced by:  tsxo1  37495  mpobi123f  37520  mptbi12f  37524  ac6s6  37530