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Theorem re1tbw1 1818
Description: tbw-ax1 1773 rederived from merco2 1809. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
re1tbw1 ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Proof of Theorem re1tbw1
StepHypRef Expression
1 mercolem8 1817 . . 3 ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))))
2 mercolem3 1812 . . 3 ((𝜓𝜒) → (𝜓 → (𝜑𝜒)))
3 mercolem6 1815 . . 3 (((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒))))) → ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))))
41, 2, 3mpsyl 68 . 2 ((𝜓𝜒) → ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒))))
5 mercolem6 1815 . 2 (((𝜓𝜒) → ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))) → ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒))))
64, 5ax-mp 5 1 ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4
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 1634  df-fal 1637
This theorem is referenced by:  re1tbw4  1821
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