Theorem cbvdisj 3976

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 2306	. . . 4 |- F/ y z e. B
3		cbvdisj.2	. . . . 5 |- F/_ x C
4	3	nfcri 2306	. . . 4 |- F/ x z e. C
5		cbvdisj.3	. . . . 5 |- ( x = y -> B = C )
6	5	eleq2d 2240	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
7	2, 4, 6	cbvrmo 2695	. . 3 |- ( E* x e. A z e. B <-> E* y e. A z e. C )
8	7	albii 1463	. 2 |- ( A. z E* x e. A z e. B <-> A. z E* y e. A z e. C )
9		df-disj 3967	. 2 |- (Disj x e. A B <-> A. z E* x e. A z e. B )
10		df-disj 3967	. 2 |- (Disj y e. A C <-> A. z E* y e. A z e. C )
11	8, 9, 10	3bitr4i 211	1 |- (Disj x e. A B <-> Disj y e. A C )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   = wceq 1348   e. wcel 2141   F/_wnfc 2299   E*wrmo 2451  Disj wdisj 3966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-reu 2455  df-rmo 2456  df-disj 3967

This theorem is referenced by:  cbvdisjv  3977  disjnims  3981