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Theorem retbwax2 1681
Description: tbw-ax2 1666 rederived from merco1 1678. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
retbwax2 (𝜑 → (𝜓𝜑))

Proof of Theorem retbwax2
StepHypRef Expression
1 merco1lem1 1679 . . . 4 (((((𝜑𝜑) → 𝜑) → (𝜑 → ⊥)) → 𝜑) → (⊥ → 𝜑))
2 merco1 1678 . . . 4 ((((((𝜑𝜑) → 𝜑) → (𝜑 → ⊥)) → 𝜑) → (⊥ → 𝜑)) → (((⊥ → 𝜑) → (𝜑𝜑)) → (𝜑 → (𝜑𝜑))))
31, 2ax-mp 5 . . 3 (((⊥ → 𝜑) → (𝜑𝜑)) → (𝜑 → (𝜑𝜑)))
4 merco1 1678 . . . 4 (((((𝜑 → (𝜑𝜑)) → (𝜑 → ⊥)) → (𝜑 → ⊥)) → ⊥) → ((⊥ → 𝜑) → (𝜑𝜑)))
5 merco1 1678 . . . 4 ((((((𝜑 → (𝜑𝜑)) → (𝜑 → ⊥)) → (𝜑 → ⊥)) → ⊥) → ((⊥ → 𝜑) → (𝜑𝜑))) → ((((⊥ → 𝜑) → (𝜑𝜑)) → (𝜑 → (𝜑𝜑))) → (𝜑 → (𝜑 → (𝜑𝜑)))))
64, 5ax-mp 5 . . 3 ((((⊥ → 𝜑) → (𝜑𝜑)) → (𝜑 → (𝜑𝜑))) → (𝜑 → (𝜑 → (𝜑𝜑))))
73, 6ax-mp 5 . 2 (𝜑 → (𝜑 → (𝜑𝜑)))
8 merco1lem1 1679 . . . 4 (((((𝜓𝜑) → 𝜑) → (𝜑 → ⊥)) → 𝜑) → (⊥ → 𝜑))
9 merco1 1678 . . . 4 ((((((𝜓𝜑) → 𝜑) → (𝜑 → ⊥)) → 𝜑) → (⊥ → 𝜑)) → (((⊥ → 𝜑) → (𝜓𝜑)) → (𝜑 → (𝜓𝜑))))
108, 9ax-mp 5 . . 3 (((⊥ → 𝜑) → (𝜓𝜑)) → (𝜑 → (𝜓𝜑)))
11 merco1 1678 . . . 4 (((((𝜑 → (𝜓𝜑)) → (𝜓 → ⊥)) → ((𝜑 → (𝜑 → (𝜑𝜑))) → ⊥)) → ⊥) → ((⊥ → 𝜑) → (𝜓𝜑)))
12 merco1 1678 . . . 4 ((((((𝜑 → (𝜓𝜑)) → (𝜓 → ⊥)) → ((𝜑 → (𝜑 → (𝜑𝜑))) → ⊥)) → ⊥) → ((⊥ → 𝜑) → (𝜓𝜑))) → ((((⊥ → 𝜑) → (𝜓𝜑)) → (𝜑 → (𝜓𝜑))) → ((𝜑 → (𝜑 → (𝜑𝜑))) → (𝜑 → (𝜓𝜑)))))
1311, 12ax-mp 5 . . 3 ((((⊥ → 𝜑) → (𝜓𝜑)) → (𝜑 → (𝜓𝜑))) → ((𝜑 → (𝜑 → (𝜑𝜑))) → (𝜑 → (𝜓𝜑))))
1410, 13ax-mp 5 . 2 ((𝜑 → (𝜑 → (𝜑𝜑))) → (𝜑 → (𝜓𝜑)))
157, 14ax-mp 5 1 (𝜑 → (𝜓𝜑))
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:  merco1lem2  1682  merco1lem3  1683  retbwax3  1688
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