Theorem ralsng 3765

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 3763	. 2 |- (A e. V -> (A.x e. {A}ph <-> [.A / x ].ph))
2		ralsng.1	. . 3 |- (x = A -> (ph <-> ps))
3	2	sbcieg 3078	. 2 |- (A e. V -> ( [.A / x ].ph <-> ps))
4	1, 3	bitrd 244	1 |- (A e. V -> (A.x e. {A}ph <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642   e. wcel 1710  A.wral 2614   [.wsbc 3046  {csn 3737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-sbc 3047  df-sn 3741

This theorem is referenced by:  ralsn  3767  ralprg  3775  raltpg  3777  ralunsn  3879  iinxsng  4042