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Theorem mdandyv4 43066
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
mdandyv4.1 (𝜑 ↔ ⊥)
mdandyv4.2 (𝜓 ↔ ⊤)
mdandyv4.3 (𝜒 ↔ ⊥)
mdandyv4.4 (𝜃 ↔ ⊥)
mdandyv4.5 (𝜏 ↔ ⊤)
mdandyv4.6 (𝜂 ↔ ⊥)
Assertion
Ref Expression
mdandyv4 ((((𝜒𝜑) ∧ (𝜃𝜑)) ∧ (𝜏𝜓)) ∧ (𝜂𝜑))

Proof of Theorem mdandyv4
StepHypRef Expression
1 mdandyv4.3 . . . . 5 (𝜒 ↔ ⊥)
2 mdandyv4.1 . . . . 5 (𝜑 ↔ ⊥)
31, 2bothfbothsame 43013 . . . 4 (𝜒𝜑)
4 mdandyv4.4 . . . . 5 (𝜃 ↔ ⊥)
54, 2bothfbothsame 43013 . . . 4 (𝜃𝜑)
63, 5pm3.2i 471 . . 3 ((𝜒𝜑) ∧ (𝜃𝜑))
7 mdandyv4.5 . . . 4 (𝜏 ↔ ⊤)
8 mdandyv4.2 . . . 4 (𝜓 ↔ ⊤)
97, 8bothtbothsame 43012 . . 3 (𝜏𝜓)
106, 9pm3.2i 471 . 2 (((𝜒𝜑) ∧ (𝜃𝜑)) ∧ (𝜏𝜓))
11 mdandyv4.6 . . 3 (𝜂 ↔ ⊥)
1211, 2bothfbothsame 43013 . 2 (𝜂𝜑)
1310, 12pm3.2i 471 1 ((((𝜒𝜑) ∧ (𝜃𝜑)) ∧ (𝜏𝜓)) ∧ (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)
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