Theorem sbcel2gv 3107

Description: Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
sbcel2gv	|- (B e. V -> ( [.B / x ].A e. x <-> A e. B))

Distinct variable group:   x,A

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

Proof of Theorem sbcel2gv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3050	. 2 |- (y = B -> ([y / x]A e. x <-> [.B / x ].A e. x))
2		eleq2 2414	. 2 |- (y = B -> (A e. y <-> A e. B))
3		nfv 1619	. . 3 |- F/x A e. y
4		eleq2 2414	. . 3 |- (x = y -> (A e. x <-> A e. y))
5	3, 4	sbie 2038	. 2 |- ([y / x]A e. x <-> A e. y)
6	1, 2, 5	vtoclbg 2916	1 |- (B e. V -> ( [.B / x ].A e. x <-> A e. B))

Syntax hints:   -> wi 4   <-> wb 176  [wsb 1648   e. wcel 1710   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by:  csbvarg  3164