Theorem ralsng 3683

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 3681	. 2 |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )
2		ralsng.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 3038	. 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 1373   e. wcel 2178   A.wral 2486   [.wsbc 3005   {csn 3643

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-sbc 3006  df-sn 3649

This theorem is referenced by:  ralsn  3686  ralprg  3694  raltpg  3696  ralunsn  3852  iinxsng  4015  posng  4765  fimax2gtrilemstep  7023  iseqf1olemqk  10689  seq3f1olemstep  10696  fimaxre2  11653  mgm1  13317  sgrp1  13358  mnd1  13402  grp1  13553  0subg  13650  ring1  13936  2sqlem10  15717  nninfsellemdc  16149