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Theorem tsim1 34262
 Description: A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
Assertion
Ref Expression
tsim1 (𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Proof of Theorem tsim1
StepHypRef Expression
1 exmid 860 . . 3 ((𝜑𝜓) ∨ ¬ (𝜑𝜓))
2 df-or 827 . . . . 5 ((¬ 𝜑𝜓) ↔ (¬ ¬ 𝜑𝜓))
3 notnotb 304 . . . . . . 7 (𝜑 ↔ ¬ ¬ 𝜑)
43bicomi 214 . . . . . 6 (¬ ¬ 𝜑𝜑)
54imbi1i 338 . . . . 5 ((¬ ¬ 𝜑𝜓) ↔ (𝜑𝜓))
62, 5bitri 264 . . . 4 ((¬ 𝜑𝜓) ↔ (𝜑𝜓))
76orbi1i 878 . . 3 (((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
81, 7mpbir 221 . 2 ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓))
98a1i 11 1 (𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 826 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 197  df-or 827 This theorem is referenced by:  mpt2bi123f  34296  ac6s6  34305
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