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 } )