Theorem cbvdisjv 4021

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 2339	. 2 |- F/_ y B
2		nfcv 2339	. 2 |- F/_ x C
3		cbvdisjv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbvdisj 4020	1 |- (Disj x e. A B <-> Disj y e. A C )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364  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: (None)