Theorem csb2 3126

Description: Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that x can be free in B but cannot occur in A. (Contributed by NM, 2-Dec-2013.)

Assertion

Ref	Expression
csb2	|- [_ A / x ]_ B = { y | E. x ( x = A /\ y e. B ) }

Distinct variable groups:   x, y, A   y, B

Allowed substitution hint:   B( x)

Proof of Theorem csb2

Step	Hyp	Ref	Expression
1		df-csb 3125	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		sbc5 3052	. . 3 |- ( [. A / x ]. y e. B <-> E. x ( x = A /\ y e. B ) )
3	2	abbii 2345	. 2 |- { y | [. A / x ]. y e. B } = { y | E. x ( x = A /\ y e. B ) }
4	1, 3	eqtri 2250	1 |- [_ A / x ]_ B = { y | E. x ( x = A /\ y e. B ) }

Syntax hints:   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   [.wsbc 3028   [_csb 3124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-csb 3125

This theorem is referenced by: (None)