Theorem sbcel21v 3096

Description: Class substitution into a membership relation. One direction of sbcel2gv 3095 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 3040	. 2 |- ( [. B / x ]. A e. x -> B e. _V )
2		sbcel2gv 3095	. . 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 2202   _Vcvv 2802   [.wsbc 3031

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032

This theorem is referenced by: (None)