Theorem sbccsb2g 3166

Description: Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)

Assertion

Ref	Expression
sbccsb2g	|- (A e. V -> ( [.A / x ].ph <-> A e. [_A / x]_{x | ph}))

Proof of Theorem sbccsb2g

Step	Hyp	Ref	Expression
1		abid 2341	. . 3 |- (x e. {x | ph} <-> ph)
2	1	sbcbii 3102	. 2 |- ( [.A / x ].x e. {x | ph} <-> [.A / x ].ph)
3		sbcel12g 3152	. . 3 |- (A e. V -> ( [.A / x ].x e. {x | ph} <-> [_A / x]_x e. [_A / x]_{x | ph}))
4		csbvarg 3164	. . . 4 |- (A e. V -> [_A / x]_x = A)
5	4	eleq1d 2419	. . 3 |- (A e. V -> ([_A / x]_x e. [_A / x]_{x | ph} <-> A e. [_A / x]_{x | ph}))
6	3, 5	bitrd 244	. 2 |- (A e. V -> ( [.A / x ].x e. {x | ph} <-> A e. [_A / x]_{x | ph}))
7	2, 6	syl5bbr 250	1 |- (A e. V -> ( [.A / x ].ph <-> A e. [_A / x]_{x | 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)