Theorem ifeq1 3560

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 2752	. . 3 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )
2	1	uneq1d 3312	. 2 |- ( A = B -> ( { x e. A | ph } u. { x e. C | -. ph } ) = ( { x e. B | ph } u. { x e. C | -. ph } ) )
3		dfif6 3559	. 2 |- if ( ph , A , C ) = ( { x e. A | ph } u. { x e. C | -. ph } )
4		dfif6 3559	. 2 |- if ( ph , B , C ) = ( { x e. B | ph } u. { x e. C | -. ph } )
5	2, 3, 4	3eqtr4g 2251	1 |- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   {crab 2476   u. cun 3151   ifcif 3557

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-un 3157  df-if 3558

This theorem is referenced by:  ifeq12  3573  ifeq1d  3574  ifbieq12i  3582  cbvsum  11503  prodeq2w  11699  cbvprod  11701  zproddc  11722