Theorem ralsns 3727

Description: Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem ralsns

Step	Hyp	Ref	Expression
1		df-ral 2525	. . 3 |- ( A. x e. { A } ph <-> A. x ( x e. { A } -> ph ) )
2		velsn 3706	. . . . 5 |- ( x e. { A } <-> x = A )
3	2	imbi1i 238	. . . 4 |- ( ( x e. { A } -> ph ) <-> ( x = A -> ph ) )
4	3	albii 1519	. . 3 |- ( A. x ( x e. { A } -> ph ) <-> A. x ( x = A -> ph ) )
5	1, 4	bitri 184	. 2 |- ( A. x e. { A } ph <-> A. x ( x = A -> ph ) )
6		sbc6g 3067	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
7	5, 6	bitr4id 199	1 |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1396   = wceq 1398   e. wcel 2203   A.wral 2520   [.wsbc 3042   {csn 3689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-sbc 3043  df-sn 3695

This theorem is referenced by:  ralsng  3729  sbcsng  3748  rabrsndc  3759  omsinds  4744  ssfirab  7197  dcfi  7268  uzsinds  10806