Theorem ifeq1 3564

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 2755	. . 3 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )
2	1	uneq1d 3316	. 2 |- ( A = B -> ( { x e. A | ph } u. { x e. C | -. ph } ) = ( { x e. B | ph } u. { x e. C | -. ph } ) )
3		dfif6 3563	. 2 |- if ( ph , A , C ) = ( { x e. A | ph } u. { x e. C | -. ph } )
4		dfif6 3563	. 2 |- if ( ph , B , C ) = ( { x e. B | ph } u. { x e. C | -. ph } )
5	2, 3, 4	3eqtr4g 2254	1 |- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   {crab 2479   u. cun 3155   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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  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-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765  df-un 3161  df-if 3562

This theorem is referenced by:  ifeq12  3577  ifeq1d  3578  ifbieq12i  3586  cbvsum  11525  prodeq2w  11721  cbvprod  11723  zproddc  11744