Theorem csbsng 3785

Description: Distribute proper substitution through the singleton of a class. csbsng 3785 is derived from the virtual deduction proof csbsngVD in set.mm. (Contributed by Alan Sare, 10-Nov-2012.)

Assertion

Ref	Expression
csbsng	|- (A e. V -> [_A / x]_{B} = {[_A / x]_B})

Proof of Theorem csbsng

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbabg 3197	. . 3 |- (A e. V -> [_A / x]_{y | y = B} = {y | [.A / x ].y = B})
2		sbceq2g 3158	. . . 4 |- (A e. V -> ( [.A / x ].y = B <-> y = [_A / x]_B))
3	2	abbidv 2467	. . 3 |- (A e. V -> {y | [.A / x ].y = B} = {y | y = [_A / x]_B})
4	1, 3	eqtrd 2385	. 2 |- (A e. V -> [_A / x]_{y | y = B} = {y | y = [_A / x]_B})
5		df-sn 3741	. . 3 |- {B} = {y | y = B}
6	5	csbeq2i 3162	. 2 |- [_A / x]_{B} = [_A / x]_{y | y = B}
7		df-sn 3741	. 2 |- {[_A / x]_B} = {y | y = [_A / x]_B}
8	4, 6, 7	3eqtr4g 2410	1 |- (A e. V -> [_A / x]_{B} = {[_A / x]_B})

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  {cab 2339   [.wsbc 3046  [_csb 3136  {csn 3737

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 2478  df-v 2861  df-sbc 3047  df-csb 3137  df-sn 3741

This theorem is referenced by: (None)