Theorem cbviun 4003

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)

Hypotheses

Ref	Expression
cbviun.1	|- F/_yB
cbviun.2	|- F/_xC
cbviun.3	|- (x = y -> B = C)

Assertion

Ref	Expression
cbviun	|- U_x e. A B = U_y e. A C

Distinct variable groups:   y,A   x,A

Allowed substitution hints:   B(x,y)   C(x,y)

Proof of Theorem cbviun

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviun.1	. . . . 5 |- F/_yB
2	1	nfcri 2483	. . . 4 |- F/y z e. B
3		cbviun.2	. . . . 5 |- F/_xC
4	3	nfcri 2483	. . . 4 |- F/x z e. C
5		cbviun.3	. . . . 5 |- (x = y -> B = C)
6	5	eleq2d 2420	. . . 4 |- (x = y -> (z e. B <-> z e. C))
7	2, 4, 6	cbvrex 2832	. . 3 |- (E.x e. A z e. B <-> E.y e. A z e. C)
8	7	abbii 2465	. 2 |- {z | E.x e. A z e. B} = {z | E.y e. A z e. C}
9		df-iun 3971	. 2 |- U_x e. A B = {z | E.x e. A z e. B}
10		df-iun 3971	. 2 |- U_y e. A C = {z | E.y e. A z e. C}
11	8, 9, 10	3eqtr4i 2383	1 |- U_x e. A B = U_y e. A C

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  {cab 2339   F/_wnfc 2476  E.wrex 2615  U_ciun 3969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-iun 3971

This theorem is referenced by:  cbviunv  4005  funiunfvf  5468  fmpt2x  5730