Theorem ifeq1 3538

Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
ifeq1	|- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )

Proof of Theorem ifeq1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabeq 2730	. . 3 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )
2	1	uneq1d 3289	. 2 |- ( A = B -> ( { x e. A | ph } u. { x e. C | -. ph } ) = ( { x e. B | ph } u. { x e. C | -. ph } ) )
3		dfif6 3537	. 2 |- if ( ph , A , C ) = ( { x e. A | ph } u. { x e. C | -. ph } )
4		dfif6 3537	. 2 |- if ( ph , B , C ) = ( { x e. B | ph } u. { x e. C | -. ph } )
5	2, 3, 4	3eqtr4g 2235	1 |- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1353   {crab 2459   u. cun 3128   ifcif 3535

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2740  df-un 3134  df-if 3536

This theorem is referenced by:  ifeq12  3551  ifeq1d  3552  ifbieq12i  3560  cbvsum  11368  prodeq2w  11564  cbvprod  11566  zproddc  11587