Theorem sbcel21v 3093

Description: Class substitution into a membership relation. One direction of sbcel2gv 3092 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Assertion

Ref	Expression
sbcel21v	|- ( [. B / x ]. A e. x -> A e. B )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem sbcel21v

Step	Hyp	Ref	Expression
1		sbcex 3037	. 2 |- ( [. B / x ]. A e. x -> B e. _V )
2		sbcel2gv 3092	. . 3 |- ( B e. _V -> ( [. B / x ]. A e. x <-> A e. B ) )
3	2	biimpd 144	. 2 |- ( B e. _V -> ( [. B / x ]. A e. x -> A e. B ) )
4	1, 3	mpcom 36	1 |- ( [. B / x ]. A e. x -> A e. B )

Syntax hints:   -> wi 4   e. wcel 2200   _Vcvv 2799   [.wsbc 3028

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029

This theorem is referenced by: (None)