Theorem sbccsbg 3165

Description: Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)

Assertion

Ref	Expression
sbccsbg	|- (A e. V -> ( [.A / x ].ph <-> y e. [_A / x]_{y | ph}))

Distinct variable group:   x,y

Allowed substitution hints:   ph(x,y)   A(x,y)   V(x,y)

Proof of Theorem sbccsbg

Step	Hyp	Ref	Expression
1		abid 2341	. . 3 |- (y e. {y | ph} <-> ph)
2	1	sbcbii 3102	. 2 |- ( [.A / x ].y e. {y | ph} <-> [.A / x ].ph)
3		sbcel2g 3158	. 2 |- (A e. V -> ( [.A / x ].y e. {y | ph} <-> y e. [_A / x]_{y | ph}))
4	2, 3	syl5bbr 250	1 |- (A e. V -> ( [.A / x ].ph <-> y e. [_A / x]_{y | ph}))

Syntax hints:   -> wi 4   <-> wb 176   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)