Theorem sbcel2gv 3041

Description: Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
sbcel2gv	|- ( B e. V -> ( [. B / x ]. A e. x <-> A e. B ) )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   V( x)

Proof of Theorem sbcel2gv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2253	. 2 |- ( x = y -> ( A e. x <-> A e. y ) )
2		eleq2 2253	. 2 |- ( y = B -> ( A e. y <-> A e. B ) )
3	1, 2	sbcie2g 3011	1 |- ( B e. V -> ( [. B / x ]. A e. x <-> A e. B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2160   [.wsbc 2977

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978

This theorem is referenced by:  sbcel21v  3042  csbvarg  3100