Theorem cbvdisjv 3949

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 2296	. 2 |- F/_ y B
2		nfcv 2296	. 2 |- F/_ x C
3		cbvdisjv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbvdisj 3948	1 |- (Disj x e. A B <-> Disj y e. A C )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332  Disj wdisj 3938

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rex 2438  df-reu 2439  df-rmo 2440  df-disj 3939

This theorem is referenced by: (None)