MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  merco1lem4 Structured version   Visualization version   GIF version

Theorem merco1lem4 1635
Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem4 (((𝜑𝜓) → 𝜒) → (𝜓𝜒))

Proof of Theorem merco1lem4
StepHypRef Expression
1 merco1lem3 1634 . . 3 ((((𝜓 → ⊥) → (𝜑 → ⊥)) → ((𝜒𝜑) → ⊥)) → ((𝜒𝜑) → (𝜓 → ⊥)))
2 merco1 1629 . . 3 (((((𝜓 → ⊥) → (𝜑 → ⊥)) → ((𝜒𝜑) → ⊥)) → ((𝜒𝜑) → (𝜓 → ⊥))) → ((((𝜒𝜑) → (𝜓 → ⊥)) → 𝜓) → (𝜑𝜓)))
31, 2ax-mp 5 . 2 ((((𝜒𝜑) → (𝜓 → ⊥)) → 𝜓) → (𝜑𝜓))
4 merco1 1629 . 2 (((((𝜒𝜑) → (𝜓 → ⊥)) → 𝜓) → (𝜑𝜓)) → (((𝜑𝜓) → 𝜒) → (𝜓𝜒)))
53, 4ax-mp 5 1 (((𝜑𝜓) → 𝜒) → (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wfal 1480
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 196  df-tru 1478  df-fal 1481
This theorem is referenced by:  merco1lem5  1636  merco1lem11  1643  merco1lem13  1645  merco1lem17  1649  merco1lem18  1650
  Copyright terms: Public domain W3C validator