Theorem tsim1 35402

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

Assertion

Ref	Expression
tsim1	⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 → 𝜓)))

Proof of Theorem tsim1

Step	Hyp	Ref	Expression
1		exmid 891	. . 3 ⊢ ((𝜑 → 𝜓) ∨ ¬ (𝜑 → 𝜓))
2		df-or 844	. . . . 5 ⊢ ((¬ 𝜑 ∨ 𝜓) ↔ (¬ ¬ 𝜑 → 𝜓))
3		notnotb 317	. . . . . . 7 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
4	3	bicomi 226	. . . . . 6 ⊢ (¬ ¬ 𝜑 ↔ 𝜑)
5	4	imbi1i 352	. . . . 5 ⊢ ((¬ ¬ 𝜑 → 𝜓) ↔ (𝜑 → 𝜓))
6	2, 5	bitri 277	. . . 4 ⊢ ((¬ 𝜑 ∨ 𝜓) ↔ (𝜑 → 𝜓))
7	6	orbi1i 910	. . 3 ⊢ (((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 → 𝜓)) ↔ ((𝜑 → 𝜓) ∨ ¬ (𝜑 → 𝜓)))
8	1, 7	mpbir 233	. 2 ⊢ ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 → 𝜓))
9	8	a1i 11	1 ⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ 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:  mpobi123f  35434  ac6s6  35444