Theorem ifiddc 3611

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 838	. 2 |- (DECID ph -> ( ph \/ -. ph ) )
2		iftrue 3580	. . 3 |- ( ph -> if ( ph , A , A ) = A )
3		iffalse 3583	. . 3 |- ( -. ph -> if ( ph , A , A ) = A )
4	2, 3	jaoi 718	. 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 710  DECID wdc 836   = wceq 1373   ifcif 3575

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-dc 837  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-if 3576

This theorem is referenced by:  xaddpnf1  9988  xaddmnf1  9990  isumz  11775  prod1dc  11972  1arithlem4  12764  xpscf  13254  lgsval2lem  15562  lgsdilem2  15588