Theorem ifeq2 3583

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 2768	. . 3 |- ( A = B -> { x e. A | -. ph } = { x e. B | -. ph } )
2	1	uneq2d 3335	. 2 |- ( A = B -> ( { x e. C | ph } u. { x e. A | -. ph } ) = ( { x e. C | ph } u. { x e. B | -. ph } ) )
3		dfif6 3581	. 2 |- if ( ph , C , A ) = ( { x e. C | ph } u. { x e. A | -. ph } )
4		dfif6 3581	. 2 |- if ( ph , C , B ) = ( { x e. C | ph } u. { x e. B | -. ph } )
5	2, 3, 4	3eqtr4g 2265	1 |- ( A = B -> if ( ph , C , A ) = if ( ph , C , B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1373   {crab 2490   u. cun 3172   ifcif 3579

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-v 2778  df-un 3178  df-if 3580

This theorem is referenced by:  ifeq12  3596  ifeq2d  3598  ifbieq2i  3603  xrmaxiflemcom  11675