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Theorem mdandyvr5 39786
Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
Hypotheses
Ref Expression
mdandyvr5.1 (𝜑𝜁)
mdandyvr5.2 (𝜓𝜎)
mdandyvr5.3 (𝜒𝜓)
mdandyvr5.4 (𝜃𝜑)
mdandyvr5.5 (𝜏𝜓)
mdandyvr5.6 (𝜂𝜑)
Assertion
Ref Expression
mdandyvr5 ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Proof of Theorem mdandyvr5
StepHypRef Expression
1 mdandyvr5.3 . . . . 5 (𝜒𝜓)
2 mdandyvr5.2 . . . . 5 (𝜓𝜎)
31, 2bitri 263 . . . 4 (𝜒𝜎)
4 mdandyvr5.4 . . . . 5 (𝜃𝜑)
5 mdandyvr5.1 . . . . 5 (𝜑𝜁)
64, 5bitri 263 . . . 4 (𝜃𝜁)
73, 6pm3.2i 470 . . 3 ((𝜒𝜎) ∧ (𝜃𝜁))
8 mdandyvr5.5 . . . 4 (𝜏𝜓)
98, 2bitri 263 . . 3 (𝜏𝜎)
107, 9pm3.2i 470 . 2 (((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜎))
11 mdandyvr5.6 . . 3 (𝜂𝜑)
1211, 5bitri 263 . 2 (𝜂𝜁)
1310, 12pm3.2i 470 1 ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383
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:  mdandyvr10  39791
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