Theorem sbcel2gv 3053

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 2260	. 2 |- ( x = y -> ( A e. x <-> A e. y ) )
2		eleq2 2260	. 2 |- ( y = B -> ( A e. y <-> A e. B ) )
3	1, 2	sbcie2g 3023	1 |- ( B e. V -> ( [. B / x ]. A e. x <-> A e. B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2167   [.wsbc 2989

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-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990

This theorem is referenced by:  sbcel21v  3054  csbvarg  3112