Theorem csb2 3139

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 3138	. 2 |- [_A / x]_B = {y | [.A / x ].y e. B}
2		sbc5 3071	. . 3 |- ( [.A / x ].y e. B <-> E.x(x = A /\ y e. B))
3	2	abbii 2466	. 2 |- {y | [.A / x ].y e. B} = {y | E.x(x = A /\ y e. B)}
4	1, 3	eqtri 2373	1 |- [_A / x]_B = {y | E.x(x = A /\ y e. B)}

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339   [.wsbc 3047  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)