Theorem ralsns 3763

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		sbc6g 3071	. 2 |- (A e. V -> ( [.A / x ].ph <-> A.x(x = A -> ph)))
2		df-ral 2619	. . 3 |- (A.x e. {A}ph <-> A.x(x e. {A} -> ph))
3		elsn 3748	. . . . 5 |- (x e. {A} <-> x = A)
4	3	imbi1i 315	. . . 4 |- ((x e. {A} -> ph) <-> (x = A -> ph))
5	4	albii 1566	. . 3 |- (A.x(x e. {A} -> ph) <-> A.x(x = A -> ph))
6	2, 5	bitri 240	. 2 |- (A.x e. {A}ph <-> A.x(x = A -> ph))
7	1, 6	syl6rbbr 255	1 |- (A e. V -> (A.x e. {A}ph <-> [.A / x ].ph))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  A.wral 2614   [.wsbc 3046  {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-ral 2619  df-v 2861  df-sbc 3047  df-sn 3741

This theorem is referenced by:  ralsng  3765  sbcsng  3783