Theorem ifiddc 3638

Description: Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)

Assertion

Ref	Expression
ifiddc	|- (DECID ph -> if ( ph , A , A ) = A )

Proof of Theorem ifiddc

Step	Hyp	Ref	Expression
1		exmiddc 841	. 2 |- (DECID ph -> ( ph \/ -. ph ) )
2		iftrue 3607	. . 3 |- ( ph -> if ( ph , A , A ) = A )
3		iffalse 3610	. . 3 |- ( -. ph -> if ( ph , A , A ) = A )
4	2, 3	jaoi 721	. 2 |- ( ( ph \/ -. ph ) -> if ( ph , A , A ) = A )
5	1, 4	syl 14	1 |- (DECID ph -> if ( ph , A , A ) = A )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 713  DECID wdc 839   = wceq 1395   ifcif 3602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-dc 840  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-if 3603

This theorem is referenced by:  xaddpnf1  10038  xaddmnf1  10040  isumz  11895  prod1dc  12092  1arithlem4  12884  xpscf  13375  lgsval2lem  15683  lgsdilem2  15709