Theorem tsbi4 38818

Description: A Tseitin axiom for logical biconditional, 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 38817	. 2 ⊢ (𝜃 → ((𝜓 ∨ ¬ 𝜑) ∨ ¬ (𝜓 ↔ 𝜑)))
2		orcom 884	. . 3 ⊢ ((𝜓 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∨ 𝜓))
3		bicom 225	. . . 4 ⊢ ((𝜓 ↔ 𝜑) ↔ (𝜑 ↔ 𝜓))
4	3	notbii 323	. . 3 ⊢ (¬ (𝜓 ↔ 𝜑) ↔ ¬ (𝜑 ↔ 𝜓))
5	2, 4	orbi12i 928	. 2 ⊢ (((𝜓 ∨ ¬ 𝜑) ∨ ¬ (𝜓 ↔ 𝜑)) ↔ ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))
6	1, 5	sylib 221	1 ⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-or 862

This theorem is used by:  tsxo4  38822  mpobi123f  38844  mptbi12f  38848  ac6s6  38854