Theorem tsbi3 33112

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 209	. . . . 5 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2		con34b 305	. . . . . 6 ⊢ ((𝜓 → 𝜑) ↔ (¬ 𝜑 → ¬ 𝜓))
3		pm2.54 388	. . . . . 6 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜑 ∨ ¬ 𝜓))
4	2, 3	sylbi 206	. . . . 5 ⊢ ((𝜓 → 𝜑) → (𝜑 ∨ ¬ 𝜓))
5	1, 4	syl 17	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ∨ ¬ 𝜓))
6	5	con3i 149	. . 3 ⊢ (¬ (𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ↔ 𝜓))
7	6	orri 390	. 2 ⊢ ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓))
8	7	a1i 11	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:  tsbi4  33113  tsxo3  33116  mpt2bi123f  33141  mptbi12f  33145  ac6s6  33150