Theorem ralsnsg 4014

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 3304	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
2		df-ral 2753	. . 3 |- ( A. x e. { A } ph <-> A. x ( x e. { A } -> ph ) )
3		elsn 3993	. . . . 5 |- ( x e. { A } <-> x = A )
4	3	imbi1i 331	. . . 4 |- ( ( x e. { A } -> ph ) <-> ( x = A -> ph ) )
5	4	albii 1701	. . 3 |- ( A. x ( x e. { A } -> ph ) <-> A. x ( x = A -> ph ) )
6	2, 5	bitri 257	. 2 |- ( A. x e. { A } ph <-> A. x ( x = A -> ph ) )
7	1, 6	syl6rbbr 272	1 |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1452   = wceq 1454   e. wcel 1897   A.wral 2748   [.wsbc 3278   {csn 3979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-ral 2753  df-v 3058  df-sbc 3279  df-sn 3980

This theorem is referenced by:  ralsng  4017  ixpsnval  7550  ac6sfi  7840  rexfiuz  13458  prmind2  14683  finixpnum  31974