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Theorem mdandyv12 39777
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
mdandyv12.1 (𝜑 ↔ ⊥)
mdandyv12.2 (𝜓 ↔ ⊤)
mdandyv12.3 (𝜒 ↔ ⊥)
mdandyv12.4 (𝜃 ↔ ⊥)
mdandyv12.5 (𝜏 ↔ ⊤)
mdandyv12.6 (𝜂 ↔ ⊤)
Assertion
Ref Expression
mdandyv12 ((((𝜒𝜑) ∧ (𝜃𝜑)) ∧ (𝜏𝜓)) ∧ (𝜂𝜓))

Proof of Theorem mdandyv12
StepHypRef Expression
1 mdandyv12.3 . . . . 5 (𝜒 ↔ ⊥)
2 mdandyv12.1 . . . . 5 (𝜑 ↔ ⊥)
31, 2bothfbothsame 39716 . . . 4 (𝜒𝜑)
4 mdandyv12.4 . . . . 5 (𝜃 ↔ ⊥)
54, 2bothfbothsame 39716 . . . 4 (𝜃𝜑)
63, 5pm3.2i 470 . . 3 ((𝜒𝜑) ∧ (𝜃𝜑))
7 mdandyv12.5 . . . 4 (𝜏 ↔ ⊤)
8 mdandyv12.2 . . . 4 (𝜓 ↔ ⊤)
97, 8bothtbothsame 39715 . . 3 (𝜏𝜓)
106, 9pm3.2i 470 . 2 (((𝜒𝜑) ∧ (𝜃𝜑)) ∧ (𝜏𝜓))
11 mdandyv12.6 . . 3 (𝜂 ↔ ⊤)
1211, 8bothtbothsame 39715 . 2 (𝜂𝜓)
1310, 12pm3.2i 470 1 ((((𝜒𝜑) ∧ (𝜃𝜑)) ∧ (𝜏𝜓)) ∧ (𝜂𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  wtru 1476  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-an 385
This theorem is referenced by: (None)
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