Theorem ralsnsg 4012

Description: Substitution expressed in terms of quantification over a singleton. (Contributed 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 3314	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
2		df-ral 2801	. . 3 |- ( A. x e. { A } ph <-> A. x ( x e. { A } -> ph ) )
3		elsn 3994	. . . . 5 |- ( x e. { A } <-> x = A )
4	3	imbi1i 325	. . . 4 |- ( ( x e. { A } -> ph ) <-> ( x = A -> ph ) )
5	4	albii 1611	. . 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 1368   = wceq 1370   e. wcel 1758   A.wral 2796   [.wsbc 3288   {csn 3980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1954  ax-ext 2431

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-ral 2801  df-v 3074  df-sbc 3289  df-sn 3981

This theorem is referenced by:  ralsng  4015  sbcsngOLD  4036  ixpsnval  7371  ac6sfi  7662  rexfiuz  12948  prmind2  13887  finixpnum  28557