Theorem ralsng 3616

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

Hypothesis

Ref	Expression
ralsng.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
ralsng	|- ( A e. V -> ( A. x e. { A } ph <-> ps ) )

Distinct variable groups:   x, A   ps, x

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

Proof of Theorem ralsng

Step	Hyp	Ref	Expression
1		ralsns 3614	. 2 |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )
2		ralsng.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 2983	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )
4	1, 3	bitrd 187	1 |- ( A e. V -> ( A. x e. { A } ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   [.wsbc 2951   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-sbc 2952  df-sn 3582

This theorem is referenced by:  ralsn  3619  ralprg  3627  raltpg  3629  ralunsn  3777  iinxsng  3939  posng  4676  fimax2gtrilemstep  6866  iseqf1olemqk  10429  seq3f1olemstep  10436  fimaxre2  11168  mgm1  12601  2sqlem10  13601  nninfsellemdc  13890