Theorem ralsng 3402

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 3399	. 2 |- (A e. V -> (A.x e. {A }ph <-> [.A / x ].ph))
2		ralsng.1	. . 3 |- (x = A -> (ph <-> ps))
3	2	sbcieg 2789	. 2 |- (A e. V -> ( [.A / x ].ph <-> ps))
4	1, 3	bitrd 177	1 |- (A e. V -> (A.x e. {A }ph <-> ps))

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1242   e. wcel 1390  A.wral 2300   [.wsbc 2758   {csn 3367

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-sbc 2759  df-sn 3373

This theorem is referenced by:  ralsn  3405  ralprg  3412  raltpg  3414  ralunsn  3559  iinxsng  3721  posng  4355