Theorem tsbi3 35415

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

Assertion

Ref	Expression
tsbi3	⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))

Proof of Theorem tsbi3

Step	Hyp	Ref	Expression
1		biimpr 222	. . . . 5 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2		con34b 318	. . . . . 6 ⊢ ((𝜓 → 𝜑) ↔ (¬ 𝜑 → ¬ 𝜓))
3		pm2.54 848	. . . . . 6 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜑 ∨ ¬ 𝜓))
4	2, 3	sylbi 219	. . . . 5 ⊢ ((𝜓 → 𝜑) → (𝜑 ∨ ¬ 𝜓))
5	1, 4	syl 17	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ∨ ¬ 𝜓))
6	5	con3i 157	. . 3 ⊢ (¬ (𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ↔ 𝜓))
7	6	orri 858	. 2 ⊢ ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓))
8	7	a1i 11	1 ⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 843

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 209  df-or 844

This theorem is referenced by:  tsbi4  35416  tsxo3  35419  mpobi123f  35442  mptbi12f  35446  ac6s6  35452