Theorem ifeq2 3575

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 2764	. . 3 |- ( A = B -> { x e. A | -. ph } = { x e. B | -. ph } )
2	1	uneq2d 3327	. 2 |- ( A = B -> ( { x e. C | ph } u. { x e. A | -. ph } ) = ( { x e. C | ph } u. { x e. B | -. ph } ) )
3		dfif6 3573	. 2 |- if ( ph , C , A ) = ( { x e. C | ph } u. { x e. A | -. ph } )
4		dfif6 3573	. 2 |- if ( ph , C , B ) = ( { x e. C | ph } u. { x e. B | -. ph } )
5	2, 3, 4	3eqtr4g 2263	1 |- ( A = B -> if ( ph , C , A ) = if ( ph , C , B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1373   {crab 2488   u. cun 3164   ifcif 3571

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774  df-un 3170  df-if 3572

This theorem is referenced by:  ifeq12  3587  ifeq2d  3589  ifbieq2i  3594  xrmaxiflemcom  11560