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Theorem ex-natded9.20-2 21679
Description: A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 21678. (Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)
Hypothesis
Ref Expression
ex-natded9.20.1  |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )
Assertion
Ref Expression
ex-natded9.20-2  |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )

Proof of Theorem ex-natded9.20-2
StepHypRef Expression
1 ex-natded9.20.1 . . . . 5  |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )
21simpld 446 . . . 4  |-  ( ph  ->  ps )
32anim1i 552 . . 3  |-  ( (
ph  /\  ch )  ->  ( ps  /\  ch ) )
43orcd 382 . 2  |-  ( (
ph  /\  ch )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
52anim1i 552 . . 3  |-  ( (
ph  /\  th )  ->  ( ps  /\  th ) )
65olcd 383 . 2  |-  ( (
ph  /\  th )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
71simprd 450 . 2  |-  ( ph  ->  ( ch  \/  th ) )
84, 6, 7mpjaodan 762 1  |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359
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 178  df-or 360  df-an 361
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