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Theorem prlem2 929
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 443 . . 3  |-  ( (
ph  /\  ps )  ->  ph )
2 simpl 443 . . 3  |-  ( ( ch  /\  th )  ->  ch )
31, 2orim12i 502 . 2  |-  ( ( ( ph  /\  ps )  \/  ( ch  /\ 
th ) )  -> 
( ph  \/  ch ) )
43pm4.71ri 614 1  |-  ( ( ( ph  /\  ps )  \/  ( ch  /\ 
th ) )  <->  ( ( ph  \/  ch )  /\  ( ( ph  /\  ps )  \/  ( ch  /\  th ) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358
This theorem is referenced by:  zfpair  4212
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 177  df-or 359  df-an 360
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