Theorem sbcel1gvOLD 37296

Description: Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 17-Aug-2018. Use sbcel1v 3338 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbcel1gvOLD	|- ( A e. V -> ( [. A / x ]. x e. B <-> A e. B ) )

Distinct variable group:   x, B

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

Proof of Theorem sbcel1gvOLD

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3282	. 2 |- ( y = A -> ( [ y / x ] x e. B <-> [. A / x ]. x e. B ) )
2		eleq1 2528	. 2 |- ( y = A -> ( y e. B <-> A e. B ) )
3		clelsb3 2568	. 2 |- ( [ y / x ] x e. B <-> y e. B )
4	1, 2, 3	vtoclbg 3120	1 |- ( A e. V -> ( [. A / x ]. x e. B <-> A e. B ) )

Syntax hints:   -> wi 4   <-> wb 189   [wsb 1808   e. wcel 1898   [.wsbc 3279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-v 3059  df-sbc 3280

This theorem is referenced by:  sbcoreleleqVD  37297  onfrALTlem4VD  37324