Theorem ifel 4129

Description: Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)

Assertion

Ref	Expression
ifel	|- ( if ( ph , A , B ) e. C <-> ( ( ph /\ A e. C ) \/ ( -. ph /\ B e. C ) ) )

Proof of Theorem ifel

Step	Hyp	Ref	Expression
1		eleq1 2689	. 2 |- ( if ( ph , A , B ) = A -> ( if ( ph , A , B ) e. C <-> A e. C ) )
2		eleq1 2689	. 2 |- ( if ( ph , A , B ) = B -> ( if ( ph , A , B ) e. C <-> B e. C ) )
3	1, 2	elimif 4122	1 |- ( if ( ph , A , B ) e. C <-> ( ( ph /\ A e. C ) \/ ( -. ph /\ B e. C ) ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   e. wcel 1990   ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by:  clwlkclwwlklem2a  26899  smatrcl  29862  clsk1independent  38344