Theorem cbvdisj 4020

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 2333	. . . 4 |- F/ y z e. B
3		cbvdisj.2	. . . . 5 |- F/_ x C
4	3	nfcri 2333	. . . 4 |- F/ x z e. C
5		cbvdisj.3	. . . . 5 |- ( x = y -> B = C )
6	5	eleq2d 2266	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
7	2, 4, 6	cbvrmo 2728	. . 3 |- ( E* x e. A z e. B <-> E* y e. A z e. C )
8	7	albii 1484	. 2 |- ( A. z E* x e. A z e. B <-> A. z E* y e. A z e. C )
9		df-disj 4011	. 2 |- (Disj x e. A B <-> A. z E* x e. A z e. B )
10		df-disj 4011	. 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 2167   F/_wnfc 2326   E*wrmo 2478  Disj wdisj 4010

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-reu 2482  df-rmo 2483  df-disj 4011

This theorem is referenced by:  cbvdisjv  4021  disjnims  4025