Theorem cbviinv 4560

Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)

Hypothesis

Ref	Expression
cbviunv.1	|- ( x = y -> B = C )

Assertion

Ref	Expression
cbviinv	|- |^|_ x e. A B = |^|_ y e. A C

Distinct variable groups:   x, A   y, A   y, B   x, C

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

Proof of Theorem cbviinv

Step	Hyp	Ref	Expression
1		nfcv 2764	. 2 |- F/_ y B
2		nfcv 2764	. 2 |- F/_ x C
3		cbviunv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbviin 4558	1 |- |^|_ x e. A B = |^|_ y e. A C

Syntax hints:   -> wi 4   = wceq 1483   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-iin 4523

This theorem is referenced by:  meaiininc  40701  iinhoiicc  40888  smflimlem3  40981  smflimlem4  40982  smflimlem6  40984  smfsuplem2  41018  smflimsuplem1  41026  smflimsup  41034