Theorem ifprdc 3819

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

Assertion

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

Proof of Theorem ifprdc

Step	Hyp	Ref	Expression
1		prid1g 3815	. . 3 |- ( A e. C -> A e. { A , B } )
2	1	3ad2ant1 1049	. 2 |- ( ( A e. C /\ B e. D /\ DECID ph ) -> A e. { A , B } )
3		prid2g 3816	. . 3 |- ( B e. D -> B e. { A , B } )
4	3	3ad2ant2 1050	. 2 |- ( ( A e. C /\ B e. D /\ DECID ph ) -> B e. { A , B } )
5		simp3 1030	. 2 |- ( ( A e. C /\ B e. D /\ DECID ph ) -> DECID ph )
6	2, 4, 5	ifcldcd 3678	1 |- ( ( A e. C /\ B e. D /\ DECID ph ) -> if ( ph , A , B ) e. { A , B } )

This proof depends on syntax axioms:   -> wi 4  DECID wdc 846   /\ w3a 1009   e. wcel 2209   ifcif 3638   {cpr 3710

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-if 3639  df-sn 3715  df-pr 3716

This theorem is used by:  indfdc  9298