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Theorem oplem1 917
Description: A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
Hypotheses
Ref Expression
oplem1.1  |-  ( ph  ->  ( ps  \/  ch ) )
oplem1.2  |-  ( ph  ->  ( th  \/  ta ) )
oplem1.3  |-  ( ps  <->  th )
oplem1.4  |-  ( ch 
->  ( th  <->  ta )
)
Assertion
Ref Expression
oplem1  |-  ( ph  ->  ps )

Proof of Theorem oplem1
StepHypRef Expression
1 oplem1.1 . 2  |-  ( ph  ->  ( ps  \/  ch ) )
2 idd 21 . . 3  |-  ( ph  ->  ( ps  ->  ps ) )
3 oplem1.2 . . . . 5  |-  ( ph  ->  ( th  \/  ta ) )
4 ax-1 5 . . . . . 6  |-  ( th 
->  ( ch  ->  th )
)
5 oplem1.4 . . . . . . 7  |-  ( ch 
->  ( th  <->  ta )
)
65biimprcd 158 . . . . . 6  |-  ( ta 
->  ( ch  ->  th )
)
74, 6jaoi 669 . . . . 5  |-  ( ( th  \/  ta )  ->  ( ch  ->  th )
)
83, 7syl 14 . . . 4  |-  ( ph  ->  ( ch  ->  th )
)
9 oplem1.3 . . . 4  |-  ( ps  <->  th )
108, 9syl6ibr 160 . . 3  |-  ( ph  ->  ( ch  ->  ps ) )
112, 10jaod 670 . 2  |-  ( ph  ->  ( ( ps  \/  ch )  ->  ps )
)
121, 11mpd 13 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> 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:  preqr1g  3566  preqr1  3568
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