Theorem sbcel1gvOLD 39094

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 3495 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 3438	. 2 |- ( y = A -> ( [ y / x ] x e. B <-> [. A / x ]. x e. B ) )
2		eleq1 2689	. 2 |- ( y = A -> ( y e. B <-> A e. B ) )
3		clelsb3 2729	. 2 |- ( [ y / x ] x e. B <-> y e. B )
4	1, 2, 3	vtoclbg 3267	1 |- ( A e. V -> ( [. A / x ]. x e. B <-> A e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   [wsb 1880   e. wcel 1990   [.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-11 2034  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:  sbcoreleleqVD  39095  onfrALTlem4VD  39122