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Theorem mdandyvr3 39784
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
mdandyvr3.1 (𝜑𝜁)
mdandyvr3.2 (𝜓𝜎)
mdandyvr3.3 (𝜒𝜓)
mdandyvr3.4 (𝜃𝜓)
mdandyvr3.5 (𝜏𝜑)
mdandyvr3.6 (𝜂𝜑)
Assertion
Ref Expression
mdandyvr3 ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Proof of Theorem mdandyvr3
StepHypRef Expression
1 mdandyvr3.3 . . . . 5 (𝜒𝜓)
2 mdandyvr3.2 . . . . 5 (𝜓𝜎)
31, 2bitri 263 . . . 4 (𝜒𝜎)
4 mdandyvr3.4 . . . . 5 (𝜃𝜓)
54, 2bitri 263 . . . 4 (𝜃𝜎)
63, 5pm3.2i 470 . . 3 ((𝜒𝜎) ∧ (𝜃𝜎))
7 mdandyvr3.5 . . . 4 (𝜏𝜑)
8 mdandyvr3.1 . . . 4 (𝜑𝜁)
97, 8bitri 263 . . 3 (𝜏𝜁)
106, 9pm3.2i 470 . 2 (((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜁))
11 mdandyvr3.6 . . 3 (𝜂𝜑)
1211, 8bitri 263 . 2 (𝜂𝜁)
1310, 12pm3.2i 470 1 ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (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:  mdandyvr12  39793
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