Theorem sbcel2gv 2967

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 2201	. 2 |- ( x = y -> ( A e. x <-> A e. y ) )
2		eleq2 2201	. 2 |- ( y = B -> ( A e. y <-> A e. B ) )
3	1, 2	sbcie2g 2937	1 |- ( B e. V -> ( [. B / x ]. A e. x <-> A e. B ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 1480   [.wsbc 2904

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905

This theorem is referenced by:  sbcel21v  2968  csbvarg  3025