Theorem cbvdisjv 3977

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 2312	. 2 |- F/_ y B
2		nfcv 2312	. 2 |- F/_ x C
3		cbvdisjv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbvdisj 3976	1 |- (Disj x e. A B <-> Disj y e. A C )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348  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: (None)