Theorem ralsnsg 4048

Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Assertion

Ref	Expression
ralsnsg	|- ( 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 ralsnsg

Step	Hyp	Ref	Expression
1		sbc6g 3350	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
2		df-ral 2809	. . 3 |- ( A. x e. { A } ph <-> A. x ( x e. { A } -> ph ) )
3		elsn 4030	. . . . 5 |- ( x e. { A } <-> x = A )
4	3	imbi1i 323	. . . 4 |- ( ( x e. { A } -> ph ) <-> ( x = A -> ph ) )
5	4	albii 1645	. . 3 |- ( A. x ( x e. { A } -> ph ) <-> A. x ( x = A -> ph ) )
6	2, 5	bitri 249	. 2 |- ( A. x e. { A } ph <-> A. x ( x = A -> ph ) )
7	1, 6	syl6rbbr 264	1 |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1396   = wceq 1398   e. wcel 1823   A.wral 2804   [.wsbc 3324   {csn 4016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-ral 2809  df-v 3108  df-sbc 3325  df-sn 4017

This theorem is referenced by:  ralsng  4051  ixpsnval  7465  ac6sfi  7756  rexfiuz  13265  prmind2  14315  finixpnum  30281