Theorem ifeq1 3608

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 2794	. . 3 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )
2	1	uneq1d 3360	. 2 |- ( A = B -> ( { x e. A | ph } u. { x e. C | -. ph } ) = ( { x e. B | ph } u. { x e. C | -. ph } ) )
3		dfif6 3607	. 2 |- if ( ph , A , C ) = ( { x e. A | ph } u. { x e. C | -. ph } )
4		dfif6 3607	. 2 |- if ( ph , B , C ) = ( { x e. B | ph } u. { x e. C | -. ph } )
5	2, 3, 4	3eqtr4g 2289	1 |- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1397   {crab 2514   u. cun 3198   ifcif 3605

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-un 3204  df-if 3606

This theorem is referenced by:  ifeq12  3622  ifeq1d  3623  ifbieq12i  3631  cbvsum  11921  prodeq2w  12118  cbvprod  12120  zproddc  12141