Theorem tsbi2 33941

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

Assertion

Ref	Expression
tsbi2	⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓)))

Proof of Theorem tsbi2

Step	Hyp	Ref	Expression
1		pm5.21 903	. . . 4 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ 𝜓))
2	1	olcd 408	. . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓)))
3		pm4.57 518	. . . . 5 ⊢ (¬ (¬ 𝜑 ∧ ¬ 𝜓) ↔ (𝜑 ∨ 𝜓))
4	3	biimpi 206	. . . 4 ⊢ (¬ (¬ 𝜑 ∧ ¬ 𝜓) → (𝜑 ∨ 𝜓))
5	4	orcd 407	. . 3 ⊢ (¬ (¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓)))
6	2, 5	pm2.61i 176	. 2 ⊢ ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓))
7	6	a1i 11	1 ⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by:  tsxo2  33945  mpt2bi123f  33971  mptbi12f  33975  ac6s6  33980