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Metamath Proof Explorer

Theorem ifpr

Description: Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007)

Ref Expression
Assertion ifpr 
|- ( ( A e. C /\ B e. D ) -> if ( ph , A , B ) e. { A , B } )

Proof

Step Hyp Ref Expression
1 elex 
 |-  ( A e. C -> A e. _V )
2 elex 
 |-  ( B e. D -> B e. _V )
3 ifcl 
 |-  ( ( A e. _V /\ B e. _V ) -> if ( ph , A , B ) e. _V )
4 ifeqor 
 |-  ( if ( ph , A , B ) = A \/ if ( ph , A , B ) = B )
5 elprg 
 |-  ( if ( ph , A , B ) e. _V -> ( if ( ph , A , B ) e. { A , B } <-> ( if ( ph , A , B ) = A \/ if ( ph , A , B ) = B ) ) )
6 4 5 mpbiri 
 |-  ( if ( ph , A , B ) e. _V -> if ( ph , A , B ) e. { A , B } )
7 3 6 syl 
 |-  ( ( A e. _V /\ B e. _V ) -> if ( ph , A , B ) e. { A , B } )
8 1 2 7 syl2an 
 |-  ( ( A e. C /\ B e. D ) -> if ( ph , A , B ) e. { A , B } )