Theorem sbcel21v 3042

Description: Class substitution into a membership relation. One direction of sbcel2gv 3041 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 2986	. 2 |- ( [. B / x ]. A e. x -> B e. _V )
2		sbcel2gv 3041	. . 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 2160   _Vcvv 2752   [.wsbc 2977

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978

This theorem is referenced by: (None)