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

Theorem barbariALT 2758
 Description: Alternate proof of barbari 2757, shorter but using more axioms. See comment of dariiALT 2754. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
barbari.maj 𝑥(𝜑𝜓)
barbari.min 𝑥(𝜒𝜑)
barbari.e 𝑥𝜒
Assertion
Ref Expression
barbariALT 𝑥(𝜒𝜓)

Proof of Theorem barbariALT
StepHypRef Expression
1 barbari.e . 2 𝑥𝜒
2 barbari.maj . . . . 5 𝑥(𝜑𝜓)
3 barbari.min . . . . 5 𝑥(𝜒𝜑)
42, 3barbara 2751 . . . 4 𝑥(𝜒𝜓)
54spi 2185 . . 3 (𝜒𝜓)
65ancli 552 . 2 (𝜒 → (𝜒𝜓))
71, 6eximii 1838 1 𝑥(𝜒𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-12 2179 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator