Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  re1tbw1 Structured version   Visualization version   GIF version

Theorem re1tbw1 1747
 Description: tbw-ax1 1702 rederived from merco2 1738. (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 1746 . . 3 ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))))
2 mercolem3 1741 . . 3 ((𝜓𝜒) → (𝜓 → (𝜑𝜒)))
3 mercolem6 1744 . . 3 (((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒))))) → ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))))
41, 2, 3mpsyl 68 . 2 ((𝜓𝜒) → ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒))))
5 mercolem6 1744 . 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 210  df-tru 1541  df-fal 1551 This theorem is referenced by:  re1tbw4  1750
 Copyright terms: Public domain W3C validator