Theorem sbcel21v 3497

Description: Class substitution into a membership relation. One direction of sbcel2gv 3496 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 3445	. 2 |- ( [. B / x ]. A e. x -> B e. _V )
2		sbcel2gv 3496	. . 3 |- ( B e. _V -> ( [. B / x ]. A e. x <-> A e. B ) )
3	2	biimpd 219	. 2 |- ( B e. _V -> ( [. B / x ]. A e. x -> A e. B ) )
4	1, 3	mpcom 38	1 |- ( [. B / x ]. A e. x -> A e. B )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by: (None)