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Theorem prlem1 928
Description: A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
Hypotheses
Ref Expression
prlem1.1  |-  ( ph  ->  ( et  <->  ch )
)
prlem1.2  |-  ( ps 
->  -.  th )
Assertion
Ref Expression
prlem1  |-  ( ph  ->  ( ps  ->  (
( ( ps  /\  ch )  \/  ( th  /\  ta ) )  ->  et ) ) )

Proof of Theorem prlem1
StepHypRef Expression
1 prlem1.1 . . . . 5  |-  ( ph  ->  ( et  <->  ch )
)
21biimprd 214 . . . 4  |-  ( ph  ->  ( ch  ->  et ) )
32adantld 453 . . 3  |-  ( ph  ->  ( ( ps  /\  ch )  ->  et ) )
4 prlem1.2 . . . . 5  |-  ( ps 
->  -.  th )
54pm2.21d 98 . . . 4  |-  ( ps 
->  ( th  ->  et ) )
65adantrd 454 . . 3  |-  ( ps 
->  ( ( th  /\  ta )  ->  et ) )
73, 6jaao 495 . 2  |-  ( (
ph  /\  ps )  ->  ( ( ( ps 
/\  ch )  \/  ( th  /\  ta ) )  ->  et ) )
87ex 423 1  |-  ( ph  ->  ( ps  ->  (
( ( ps  /\  ch )  \/  ( th  /\  ta ) )  ->  et ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> 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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