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Theorem mdandyv1 43063
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
mdandyv1.1 (𝜑 ↔ ⊥)
mdandyv1.2 (𝜓 ↔ ⊤)
mdandyv1.3 (𝜒 ↔ ⊤)
mdandyv1.4 (𝜃 ↔ ⊥)
mdandyv1.5 (𝜏 ↔ ⊥)
mdandyv1.6 (𝜂 ↔ ⊥)
Assertion
Ref Expression
mdandyv1 ((((𝜒𝜓) ∧ (𝜃𝜑)) ∧ (𝜏𝜑)) ∧ (𝜂𝜑))

Proof of Theorem mdandyv1
StepHypRef Expression
1 mdandyv1.3 . . . . 5 (𝜒 ↔ ⊤)
2 mdandyv1.2 . . . . 5 (𝜓 ↔ ⊤)
31, 2bothtbothsame 43012 . . . 4 (𝜒𝜓)
4 mdandyv1.4 . . . . 5 (𝜃 ↔ ⊥)
5 mdandyv1.1 . . . . 5 (𝜑 ↔ ⊥)
64, 5bothfbothsame 43013 . . . 4 (𝜃𝜑)
73, 6pm3.2i 471 . . 3 ((𝜒𝜓) ∧ (𝜃𝜑))
8 mdandyv1.5 . . . 4 (𝜏 ↔ ⊥)
98, 5bothfbothsame 43013 . . 3 (𝜏𝜑)
107, 9pm3.2i 471 . 2 (((𝜒𝜓) ∧ (𝜃𝜑)) ∧ (𝜏𝜑))
11 mdandyv1.6 . . 3 (𝜂 ↔ ⊥)
1211, 5bothfbothsame 43013 . 2 (𝜂𝜑)
1310, 12pm3.2i 471 1 ((((𝜒𝜓) ∧ (𝜃𝜑)) ∧ (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)
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