Theorem ralsng 3564

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 3562	. 2 |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )
2		ralsng.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 2941	. 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 1331   e. wcel 1480   A.wral 2416   [.wsbc 2909   {csn 3527

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-sbc 2910  df-sn 3533

This theorem is referenced by:  ralsn  3567  ralprg  3574  raltpg  3576  ralunsn  3724  iinxsng  3886  posng  4611  fimax2gtrilemstep  6794  iseqf1olemqk  10267  seq3f1olemstep  10274  fimaxre2  10998  nninfsellemdc  13206