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Theorem merco1lem1 1679
Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem1 (𝜑 → (⊥ → 𝜒))

Proof of Theorem merco1lem1
StepHypRef Expression
1 merco1 1678 . . . . 5 (((((⊥ → 𝜑) → (𝜑 → ⊥)) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (((⊥ → 𝜑) → ⊥) → (𝜑 → ⊥)))
2 merco1 1678 . . . . 5 ((((((⊥ → 𝜑) → (𝜑 → ⊥)) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (((⊥ → 𝜑) → ⊥) → (𝜑 → ⊥))) → (((((⊥ → 𝜑) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))))
31, 2ax-mp 5 . . . 4 (((((⊥ → 𝜑) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑)))
4 merco1 1678 . . . 4 ((((((⊥ → 𝜑) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))) → (((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))))
53, 4ax-mp 5 . . 3 (((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑)))
6 merco1 1678 . . . . 5 (((((⊥ → 𝜑) → (𝜑 → ⊥)) → ((𝜑 → (⊥ → 𝜑)) → ⊥)) → (𝜑 → (⊥ → 𝜑))) → (((𝜑 → (⊥ → 𝜑)) → ⊥) → (𝜑 → ⊥)))
7 merco1 1678 . . . . 5 ((((((⊥ → 𝜑) → (𝜑 → ⊥)) → ((𝜑 → (⊥ → 𝜑)) → ⊥)) → (𝜑 → (⊥ → 𝜑))) → (((𝜑 → (⊥ → 𝜑)) → ⊥) → (𝜑 → ⊥))) → (((((𝜑 → (⊥ → 𝜑)) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → ((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑))))
86, 7ax-mp 5 . . . 4 (((((𝜑 → (⊥ → 𝜑)) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → ((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑)))
9 merco1 1678 . . . 4 ((((((𝜑 → (⊥ → 𝜑)) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → ((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑))) → ((((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))) → (𝜑 → (𝜑 → (⊥ → 𝜑)))))
108, 9ax-mp 5 . . 3 ((((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))) → (𝜑 → (𝜑 → (⊥ → 𝜑))))
115, 10ax-mp 5 . 2 (𝜑 → (𝜑 → (⊥ → 𝜑)))
12 merco1 1678 . . . . 5 (((((⊥ → 𝜑) → (𝜑 → ⊥)) → (𝜑 → ⊥)) → (⊥ → 𝜒)) → (((⊥ → 𝜒) → ⊥) → (𝜑 → ⊥)))
13 merco1 1678 . . . . 5 ((((((⊥ → 𝜑) → (𝜑 → ⊥)) → (𝜑 → ⊥)) → (⊥ → 𝜒)) → (((⊥ → 𝜒) → ⊥) → (𝜑 → ⊥))) → (((((⊥ → 𝜒) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))))
1412, 13ax-mp 5 . . . 4 (((((⊥ → 𝜒) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑)))
15 merco1 1678 . . . 4 ((((((⊥ → 𝜒) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))) → (((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒)) → (𝜑 → (⊥ → 𝜒))))
1614, 15ax-mp 5 . . 3 (((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒)) → (𝜑 → (⊥ → 𝜒)))
17 merco1 1678 . . . . 5 (((((⊥ → 𝜒) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥)) → ((𝜑 → (⊥ → 𝜑)) → ⊥)) → (𝜑 → (⊥ → 𝜒))) → (((𝜑 → (⊥ → 𝜒)) → ⊥) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥)))
18 merco1 1678 . . . . 5 ((((((⊥ → 𝜒) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥)) → ((𝜑 → (⊥ → 𝜑)) → ⊥)) → (𝜑 → (⊥ → 𝜒))) → (((𝜑 → (⊥ → 𝜒)) → ⊥) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥))) → (((((𝜑 → (⊥ → 𝜒)) → ⊥) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥)) → (⊥ → 𝜒)) → ((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒))))
1917, 18ax-mp 5 . . . 4 (((((𝜑 → (⊥ → 𝜒)) → ⊥) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥)) → (⊥ → 𝜒)) → ((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒)))
20 merco1 1678 . . . 4 ((((((𝜑 → (⊥ → 𝜒)) → ⊥) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥)) → (⊥ → 𝜒)) → ((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒))) → ((((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒)) → (𝜑 → (⊥ → 𝜒))) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → (𝜑 → (⊥ → 𝜒)))))
2119, 20ax-mp 5 . . 3 ((((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒)) → (𝜑 → (⊥ → 𝜒))) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → (𝜑 → (⊥ → 𝜒))))
2216, 21ax-mp 5 . 2 ((𝜑 → (𝜑 → (⊥ → 𝜑))) → (𝜑 → (⊥ → 𝜒)))
2311, 22ax-mp 5 1 (𝜑 → (⊥ → 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wfal 1528
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-tru 1526  df-fal 1529
This theorem is referenced by:  retbwax4  1680  retbwax2  1681
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