Theorem ifeq2 3473

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

Assertion

Ref	Expression
ifeq2	|- ( A = B -> if ( ph , C , A ) = if ( ph , C , B ) )

Proof of Theorem ifeq2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabeq 2673	. . 3 |- ( A = B -> { x e. A | -. ph } = { x e. B | -. ph } )
2	1	uneq2d 3225	. 2 |- ( A = B -> ( { x e. C | ph } u. { x e. A | -. ph } ) = ( { x e. C | ph } u. { x e. B | -. ph } ) )
3		dfif6 3471	. 2 |- if ( ph , C , A ) = ( { x e. C | ph } u. { x e. A | -. ph } )
4		dfif6 3471	. 2 |- if ( ph , C , B ) = ( { x e. C | ph } u. { x e. B | -. ph } )
5	2, 3, 4	3eqtr4g 2195	1 |- ( A = B -> if ( ph , C , A ) = if ( ph , C , B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1331   {crab 2418   u. cun 3064   ifcif 3469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-v 2683  df-un 3070  df-if 3470

This theorem is referenced by:  ifeq12  3483  ifeq2d  3485  ifbieq2i  3490  xrmaxiflemcom  11011