Theorem sbcel21v 2903

Description: Class substitution into a membership relation. One direction of sbcel2gv 2902 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 2848	. 2 |- ( [. B / x ]. A e. x -> B e. _V )
2		sbcel2gv 2902	. . 3 |- ( B e. _V -> ( [. B / x ]. A e. x <-> A e. B ) )
3	2	biimpd 142	. 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 1438   _Vcvv 2619   [.wsbc 2840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841

This theorem is referenced by: (None)