Theorem cbvdisj 3991

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 2313	. . . 4 |- F/ y z e. B
3		cbvdisj.2	. . . . 5 |- F/_ x C
4	3	nfcri 2313	. . . 4 |- F/ x z e. C
5		cbvdisj.3	. . . . 5 |- ( x = y -> B = C )
6	5	eleq2d 2247	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
7	2, 4, 6	cbvrmo 2703	. . 3 |- ( E* x e. A z e. B <-> E* y e. A z e. C )
8	7	albii 1470	. 2 |- ( A. z E* x e. A z e. B <-> A. z E* y e. A z e. C )
9		df-disj 3982	. 2 |- (Disj x e. A B <-> A. z E* x e. A z e. B )
10		df-disj 3982	. 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 1351   = wceq 1353   e. wcel 2148   F/_wnfc 2306   E*wrmo 2458  Disj wdisj 3981

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-reu 2462  df-rmo 2463  df-disj 3982

This theorem is referenced by:  cbvdisjv  3992  disjnims  3996