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Theorem mdandyv8 43070
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
mdandyv8.1 (𝜑 ↔ ⊥)
mdandyv8.2 (𝜓 ↔ ⊤)
mdandyv8.3 (𝜒 ↔ ⊥)
mdandyv8.4 (𝜃 ↔ ⊥)
mdandyv8.5 (𝜏 ↔ ⊥)
mdandyv8.6 (𝜂 ↔ ⊤)
Assertion
Ref Expression
mdandyv8 ((((𝜒𝜑) ∧ (𝜃𝜑)) ∧ (𝜏𝜑)) ∧ (𝜂𝜓))

Proof of Theorem mdandyv8
StepHypRef Expression
1 mdandyv8.3 . . . . 5 (𝜒 ↔ ⊥)
2 mdandyv8.1 . . . . 5 (𝜑 ↔ ⊥)
31, 2bothfbothsame 43013 . . . 4 (𝜒𝜑)
4 mdandyv8.4 . . . . 5 (𝜃 ↔ ⊥)
54, 2bothfbothsame 43013 . . . 4 (𝜃𝜑)
63, 5pm3.2i 471 . . 3 ((𝜒𝜑) ∧ (𝜃𝜑))
7 mdandyv8.5 . . . 4 (𝜏 ↔ ⊥)
87, 2bothfbothsame 43013 . . 3 (𝜏𝜑)
96, 8pm3.2i 471 . 2 (((𝜒𝜑) ∧ (𝜃𝜑)) ∧ (𝜏𝜑))
10 mdandyv8.6 . . 3 (𝜂 ↔ ⊤)
11 mdandyv8.2 . . 3 (𝜓 ↔ ⊤)
1210, 11bothtbothsame 43012 . 2 (𝜂𝜓)
139, 12pm3.2i 471 1 ((((𝜒𝜑) ∧ (𝜃𝜑)) ∧ (𝜏𝜑)) ∧ (𝜂𝜓))
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)
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