MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  merco1lem5 Structured version   Visualization version   GIF version

Theorem merco1lem5 1685
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
merco1lem5 ((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑𝜏))

Proof of Theorem merco1lem5
StepHypRef Expression
1 merco1lem4 1684 . 2 ((((𝜏𝜑) → (𝜑 → ⊥)) → 𝜒) → ((𝜑 → ⊥) → 𝜒))
2 merco1 1678 . 2 (((((𝜏𝜑) → (𝜑 → ⊥)) → 𝜒) → ((𝜑 → ⊥) → 𝜒)) → ((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑𝜏)))
31, 2ax-mp 5 1 ((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑𝜏))
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:  merco1lem6  1686  merco1lem7  1687  merco1lem11  1692  merco1lem18  1699
  Copyright terms: Public domain W3C validator