Theorem sbcel1g 3021

Description: Move proper substitution in and out of a membership relation. Note that the scope of [. A / x ]. is the wff B e. C, whereas the scope of [_ A / x ]_ is the class B. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
sbcel1g	|- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. C ) )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)   V( x)

Proof of Theorem sbcel1g

Step	Hyp	Ref	Expression
1		sbcel12g 3017	. 2 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
2		csbconstg 3016	. . 3 |- ( A e. V -> [_ A / x ]_ C = C )
3	2	eleq2d 2209	. 2 |- ( A e. V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> [_ A / x ]_ B e. C ) )
4	1, 3	bitrd 187	1 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. C ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 1480   [.wsbc 2909   [_csb 3003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004

This theorem is referenced by:  rspcsbela  3059