Theorem csb2 3000

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 2999	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		sbc5 2927	. . 3 |- ( [. A / x ]. y e. B <-> E. x ( x = A /\ y e. B ) )
3	2	abbii 2253	. 2 |- { y | [. A / x ]. y e. B } = { y | E. x ( x = A /\ y e. B ) }
4	1, 3	eqtri 2158	1 |- [_ A / x ]_ B = { y | E. x ( x = A /\ y e. B ) }

Syntax hints:   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2123   [.wsbc 2904   [_csb 2998

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 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905  df-csb 2999

This theorem is referenced by: (None)