Theorem ralsng 3706

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 3704	. 2 |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )
2		ralsng.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 3061	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )
4	1, 3	bitrd 188	1 |- ( A e. V -> ( A. x e. { A } ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   [.wsbc 3028   {csn 3666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-sbc 3029  df-sn 3672

This theorem is referenced by:  ralsn  3709  ralprg  3717  raltpg  3719  ralunsn  3875  iinxsng  4038  posng  4790  fimax2gtrilemstep  7058  iseqf1olemqk  10724  seq3f1olemstep  10731  fimaxre2  11733  mgm1  13398  sgrp1  13439  mnd1  13483  grp1  13634  0subg  13731  ring1  14017  2sqlem10  15798  nninfsellemdc  16335