Theorem csb2 3051

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 3050	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		sbc5 2978	. . 3 |- ( [. A / x ]. y e. B <-> E. x ( x = A /\ y e. B ) )
3	2	abbii 2286	. 2 |- { y | [. A / x ]. y e. B } = { y | E. x ( x = A /\ y e. B ) }
4	1, 3	eqtri 2191	1 |- [_ A / x ]_ B = { y | E. x ( x = A /\ y e. B ) }

Syntax hints:   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   [.wsbc 2955   [_csb 3049

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by: (None)