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Theorem prlem2 916
Description: A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
Assertion
Ref Expression
prlem2  |-  ( ( ( ph  /\  ps )  \/  ( ch  /\ 
th ) )  <->  ( ( ph  \/  ch )  /\  ( ( ph  /\  ps )  \/  ( ch  /\  th ) ) ) )

Proof of Theorem prlem2
StepHypRef Expression
1 simpl 107 . . 3  |-  ( (
ph  /\  ps )  ->  ph )
2 simpl 107 . . 3  |-  ( ( ch  /\  th )  ->  ch )
31, 2orim12i 709 . 2  |-  ( ( ( ph  /\  ps )  \/  ( ch  /\ 
th ) )  -> 
( ph  \/  ch ) )
43pm4.71ri 384 1  |-  ( ( ( ph  /\  ps )  \/  ( ch  /\ 
th ) )  <->  ( ( ph  \/  ch )  /\  ( ( ph  /\  ps )  \/  ( ch  /\  th ) ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    \/ wo 662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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