Users' Mathboxes Mathbox for Jarvin Udandy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mdandyv2 Structured version   Visualization version   GIF version

Theorem mdandyv2 43064
Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
Hypotheses
Ref Expression
mdandyv2.1 (𝜑 ↔ ⊥)
mdandyv2.2 (𝜓 ↔ ⊤)
mdandyv2.3 (𝜒 ↔ ⊥)
mdandyv2.4 (𝜃 ↔ ⊤)
mdandyv2.5 (𝜏 ↔ ⊥)
mdandyv2.6 (𝜂 ↔ ⊥)
Assertion
Ref Expression
mdandyv2 ((((𝜒𝜑) ∧ (𝜃𝜓)) ∧ (𝜏𝜑)) ∧ (𝜂𝜑))

Proof of Theorem mdandyv2
StepHypRef Expression
1 mdandyv2.3 . . . . 5 (𝜒 ↔ ⊥)
2 mdandyv2.1 . . . . 5 (𝜑 ↔ ⊥)
31, 2bothfbothsame 43013 . . . 4 (𝜒𝜑)
4 mdandyv2.4 . . . . 5 (𝜃 ↔ ⊤)
5 mdandyv2.2 . . . . 5 (𝜓 ↔ ⊤)
64, 5bothtbothsame 43012 . . . 4 (𝜃𝜓)
73, 6pm3.2i 471 . . 3 ((𝜒𝜑) ∧ (𝜃𝜓))
8 mdandyv2.5 . . . 4 (𝜏 ↔ ⊥)
98, 2bothfbothsame 43013 . . 3 (𝜏𝜑)
107, 9pm3.2i 471 . 2 (((𝜒𝜑) ∧ (𝜃𝜓)) ∧ (𝜏𝜑))
11 mdandyv2.6 . . 3 (𝜂 ↔ ⊥)
1211, 2bothfbothsame 43013 . 2 (𝜂𝜑)
1310, 12pm3.2i 471 1 ((((𝜒𝜑) ∧ (𝜃𝜓)) ∧ (𝜏𝜑)) ∧ (𝜂𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wtru 1529  wfal 1540
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 208  df-an 397
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator