Theorem cbvdisjv 3970

Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)

Hypothesis

Ref	Expression
cbvdisjv.1	|- ( x = y -> B = C )

Assertion

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

Distinct variable groups:   x, y, A   y, B   x, C

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

Proof of Theorem cbvdisjv

Step	Hyp	Ref	Expression
1		nfcv 2308	. 2 |- F/_ y B
2		nfcv 2308	. 2 |- F/_ x C
3		cbvdisjv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbvdisj 3969	1 |- (Disj x e. A B <-> Disj y e. A C )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343  Disj wdisj 3959

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-reu 2451  df-rmo 2452  df-disj 3960

This theorem is referenced by: (None)