Theorem ralsng 3571

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 3569	. 2 |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )
2		ralsng.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 2945	. 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 1332   e. wcel 1481   A.wral 2417   [.wsbc 2913   {csn 3532

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-sbc 2914  df-sn 3538

This theorem is referenced by:  ralsn  3574  ralprg  3582  raltpg  3584  ralunsn  3732  iinxsng  3894  posng  4619  fimax2gtrilemstep  6802  iseqf1olemqk  10298  seq3f1olemstep  10305  fimaxre2  11030  nninfsellemdc  13381