Theorem csb2 3074

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 3073	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		sbc5 3001	. . 3 |- ( [. A / x ]. y e. B <-> E. x ( x = A /\ y e. B ) )
3	2	abbii 2305	. 2 |- { y | [. A / x ]. y e. B } = { y | E. x ( x = A /\ y e. B ) }
4	1, 3	eqtri 2210	1 |- [_ A / x ]_ B = { y | E. x ( x = A /\ y e. B ) }

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2160   {cab 2175   [.wsbc 2977   [_csb 3072

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978  df-csb 3073

This theorem is referenced by: (None)