Theorem ifiddc 3505

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 821	. 2 |- (DECID ph -> ( ph \/ -. ph ) )
2		iftrue 3479	. . 3 |- ( ph -> if ( ph , A , A ) = A )
3		iffalse 3482	. . 3 |- ( -. ph -> if ( ph , A , A ) = A )
4	2, 3	jaoi 705	. 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 697  DECID wdc 819   = wceq 1331   ifcif 3474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-dc 820  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-if 3475

This theorem is referenced by:  xaddpnf1  9629  xaddmnf1  9631  isumz  11158