Theorem ifiddc 3595

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 837	. 2 |- (DECID ph -> ( ph \/ -. ph ) )
2		iftrue 3566	. . 3 |- ( ph -> if ( ph , A , A ) = A )
3		iffalse 3569	. . 3 |- ( -. ph -> if ( ph , A , A ) = A )
4	2, 3	jaoi 717	. 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 709  DECID wdc 835   = wceq 1364   ifcif 3561

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 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-dc 836  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-if 3562

This theorem is referenced by:  xaddpnf1  9921  xaddmnf1  9923  isumz  11554  prod1dc  11751  1arithlem4  12535  xpscf  12990  lgsval2lem  15251  lgsdilem2  15277