Theorem ifiddc 3641

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 843	. 2 |- (DECID ph -> ( ph \/ -. ph ) )
2		iftrue 3610	. . 3 |- ( ph -> if ( ph , A , A ) = A )
3		iffalse 3613	. . 3 |- ( -. ph -> if ( ph , A , A ) = A )
4	2, 3	jaoi 723	. 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 715  DECID wdc 841   = wceq 1397   ifcif 3605

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 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 842  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-if 3606

This theorem is referenced by:  xaddpnf1  10080  xaddmnf1  10082  isumz  11949  prod1dc  12146  1arithlem4  12938  xpscf  13429  lgsval2lem  15738  lgsdilem2  15764