Theorem cbvdisj 4005

Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
cbvdisj.1	|- F/_ y B
cbvdisj.2	|- F/_ x C
cbvdisj.3	|- ( x = y -> B = C )

Assertion

Ref	Expression
cbvdisj	|- (Disj x e. A B <-> Disj y e. A C )

Distinct variable group:   x, y, A

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

Proof of Theorem cbvdisj

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvdisj.1	. . . . 5 |- F/_ y B
2	1	nfcri 2326	. . . 4 |- F/ y z e. B
3		cbvdisj.2	. . . . 5 |- F/_ x C
4	3	nfcri 2326	. . . 4 |- F/ x z e. C
5		cbvdisj.3	. . . . 5 |- ( x = y -> B = C )
6	5	eleq2d 2259	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
7	2, 4, 6	cbvrmo 2717	. . 3 |- ( E* x e. A z e. B <-> E* y e. A z e. C )
8	7	albii 1481	. 2 |- ( A. z E* x e. A z e. B <-> A. z E* y e. A z e. C )
9		df-disj 3996	. 2 |- (Disj x e. A B <-> A. z E* x e. A z e. B )
10		df-disj 3996	. 2 |- (Disj y e. A C <-> A. z E* y e. A z e. C )
11	8, 9, 10	3bitr4i 212	1 |- (Disj x e. A B <-> Disj y e. A C )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2160   F/_wnfc 2319   E*wrmo 2471  Disj wdisj 3995

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-reu 2475  df-rmo 2476  df-disj 3996

This theorem is referenced by:  cbvdisjv  4006  disjnims  4010