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Theorem mdandyvr4 41457
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
mdandyvr4.1 (𝜑𝜁)
mdandyvr4.2 (𝜓𝜎)
mdandyvr4.3 (𝜒𝜑)
mdandyvr4.4 (𝜃𝜑)
mdandyvr4.5 (𝜏𝜓)
mdandyvr4.6 (𝜂𝜑)
Assertion
Ref Expression
mdandyvr4 ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Proof of Theorem mdandyvr4
StepHypRef Expression
1 mdandyvr4.3 . . . . 5 (𝜒𝜑)
2 mdandyvr4.1 . . . . 5 (𝜑𝜁)
31, 2bitri 264 . . . 4 (𝜒𝜁)
4 mdandyvr4.4 . . . . 5 (𝜃𝜑)
54, 2bitri 264 . . . 4 (𝜃𝜁)
63, 5pm3.2i 470 . . 3 ((𝜒𝜁) ∧ (𝜃𝜁))
7 mdandyvr4.5 . . . 4 (𝜏𝜓)
8 mdandyvr4.2 . . . 4 (𝜓𝜎)
97, 8bitri 264 . . 3 (𝜏𝜎)
106, 9pm3.2i 470 . 2 (((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜎))
11 mdandyvr4.6 . . 3 (𝜂𝜑)
1211, 2bitri 264 . 2 (𝜂𝜁)
1310, 12pm3.2i 470 1 ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))
Colors of variables: wff setvar class
Syntax hints:  wb 196  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 197  df-an 385
This theorem is referenced by:  mdandyvr11  41464
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