Theorem tsbi4 33113

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

Assertion

Ref	Expression
tsbi4	⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))

Proof of Theorem tsbi4

Step	Hyp	Ref	Expression
1		tsbi3 33112	. 2 ⊢ (𝜃 → ((𝜓 ∨ ¬ 𝜑) ∨ ¬ (𝜓 ↔ 𝜑)))
2		orcom 401	. . 3 ⊢ ((𝜓 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∨ 𝜓))
3		bicom 211	. . . 4 ⊢ ((𝜓 ↔ 𝜑) ↔ (𝜑 ↔ 𝜓))
4	3	notbii 309	. . 3 ⊢ (¬ (𝜓 ↔ 𝜑) ↔ ¬ (𝜑 ↔ 𝜓))
5	2, 4	orbi12i 542	. 2 ⊢ (((𝜓 ∨ ¬ 𝜑) ∨ ¬ (𝜓 ↔ 𝜑)) ↔ ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))
6	1, 5	sylib 207	1 ⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382

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

This theorem is referenced by:  tsxo4  33117  mpt2bi123f  33141  mptbi12f  33145  ac6s6  33150