Theorem ifiddc 3676

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 848	. 2 |- (DECID ph -> ( ph \/ -. ph ) )
2		iftrue 3645	. . 3 |- ( ph -> if ( ph , A , A ) = A )
3		iffalse 3648	. . 3 |- ( -. ph -> if ( ph , A , A ) = A )
4	2, 3	jaoi 728	. 2 |- ( ( ph \/ -. ph ) -> if ( ph , A , A ) = A )
5	1, 4	syl 14	1 |- (DECID ph -> if ( ph , A , A ) = A )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   \/ wo 720  DECID wdc 846   = wceq 1402   ifcif 3638

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-dc 847  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-if 3639

This theorem is used by:  xaddpnf1  10248  xaddmnf1  10250  isumz  12156  prod1dc  12353  1arithlem4  13145  xpscf  13668  lgsval2lem  16129  lgsdilem2  16155