Theorem cbvdisj 4031

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 2342	. . . 4 |- F/ y z e. B
3		cbvdisj.2	. . . . 5 |- F/_ x C
4	3	nfcri 2342	. . . 4 |- F/ x z e. C
5		cbvdisj.3	. . . . 5 |- ( x = y -> B = C )
6	5	eleq2d 2275	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
7	2, 4, 6	cbvrmo 2737	. . 3 |- ( E* x e. A z e. B <-> E* y e. A z e. C )
8	7	albii 1493	. 2 |- ( A. z E* x e. A z e. B <-> A. z E* y e. A z e. C )
9		df-disj 4022	. 2 |- (Disj x e. A B <-> A. z E* x e. A z e. B )
10		df-disj 4022	. 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 1371   = wceq 1373   e. wcel 2176   F/_wnfc 2335   E*wrmo 2487  Disj wdisj 4021

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-reu 2491  df-rmo 2492  df-disj 4022

This theorem is referenced by:  cbvdisjv  4032  disjnims  4036