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Theorem prth 570
Description: Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it 'praeclarum theorema' (splendid theorem). (The proof was shortened by Wolf Lammen, 7-Apr-2013.)
Assertion
Ref Expression
prth |- (((ph -> ps) /\ (ch -> th)) -> ((ph /\ ch) -> (ps /\ th)))

Proof of Theorem prth
StepHypRef Expression
1 simpl 459 . 2 |- (((ph -> ps) /\ (ch -> th)) -> (ph -> ps))
2 simpr 463 . 2 |- (((ph -> ps) /\ (ch -> th)) -> (ch -> th))
31, 2anim12d 563 1 |- (((ph -> ps) /\ (ch -> th)) -> ((ph /\ ch) -> (ps /\ th)))
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 377
This theorem is referenced by:  mo 1865  2mo 1922  euind 2512  reuind 2523  reuss2 2915  opelopabt 3596  reusv3i 3851  tfrlem5 5339  spanuni 13389  pm11.71 17417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 185  df-an 379
Copyright terms: Public domain