Theorem csb2 3535

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 3534	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		sbc5 3460	. . 3 |- ( [. A / x ]. y e. B <-> E. x ( x = A /\ y e. B ) )
3	2	abbii 2739	. 2 |- { y | [. A / x ]. y e. B } = { y | E. x ( x = A /\ y e. B ) }
4	1, 3	eqtri 2644	1 |- [_ A / x ]_ B = { y | E. x ( x = A /\ y e. B ) }

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by: (None)