Users' Mathboxes Mathbox for Jarvin Udandy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mdandyvr2 Structured version   Visualization version   GIF version

Theorem mdandyvr2 39783
Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
Hypotheses
Ref Expression
mdandyvr2.1 (𝜑𝜁)
mdandyvr2.2 (𝜓𝜎)
mdandyvr2.3 (𝜒𝜑)
mdandyvr2.4 (𝜃𝜓)
mdandyvr2.5 (𝜏𝜑)
mdandyvr2.6 (𝜂𝜑)
Assertion
Ref Expression
mdandyvr2 ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Proof of Theorem mdandyvr2
StepHypRef Expression
1 mdandyvr2.3 . . . . 5 (𝜒𝜑)
2 mdandyvr2.1 . . . . 5 (𝜑𝜁)
31, 2bitri 263 . . . 4 (𝜒𝜁)
4 mdandyvr2.4 . . . . 5 (𝜃𝜓)
5 mdandyvr2.2 . . . . 5 (𝜓𝜎)
64, 5bitri 263 . . . 4 (𝜃𝜎)
73, 6pm3.2i 470 . . 3 ((𝜒𝜁) ∧ (𝜃𝜎))
8 mdandyvr2.5 . . . 4 (𝜏𝜑)
98, 2bitri 263 . . 3 (𝜏𝜁)
107, 9pm3.2i 470 . 2 (((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜁))
11 mdandyvr2.6 . . 3 (𝜂𝜑)
1211, 2bitri 263 . 2 (𝜂𝜁)
1310, 12pm3.2i 470 1 ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383
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-an 385
This theorem is referenced by:  mdandyvr13  39794
  Copyright terms: Public domain W3C validator