Theorem ifeq1 3621

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 2804	. . 3 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )
2	1	uneq1d 3371	. 2 |- ( A = B -> ( { x e. A | ph } u. { x e. C | -. ph } ) = ( { x e. B | ph } u. { x e. C | -. ph } ) )
3		dfif6 3618	. 2 |- if ( ph , A , C ) = ( { x e. A | ph } u. { x e. C | -. ph } )
4		dfif6 3618	. 2 |- if ( ph , B , C ) = ( { x e. B | ph } u. { x e. C | -. ph } )
5	2, 3, 4	3eqtr4g 2290	1 |- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   {crab 2524   u. cun 3208   ifcif 3616

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2814  df-un 3214  df-if 3617

This theorem is referenced by:  ifeq12  3635  ifeq1d  3636  ifbieq12i  3644  cbvsum  12023  prodeq2w  12220  cbvprod  12222  zproddc  12243