Theorem ralxfr2d 4668

Description: Transfer universal quantification from a variable to another variable contained in expression . (Contributed by Mario Carneiro, 20-Aug-2014.)

Hypotheses

Ref	Expression
ralxfr2d.1
ralxfr2d.2
ralxfr2d.3

Assertion

Ref	Expression
ralxfr2d

Distinct variable groups:   ,   ,,   ,   ,   ,,   ,

Proof of Theorem ralxfr2d

Step	Hyp	Ref	Expression
1		ralxfr2d.1	. . . 4
2		elisset 3120	. . . 4
3	1, 2	syl 16	. . 3
4		ralxfr2d.2	. . . . . . . 8
5	4	biimprd 223	. . . . . . 7
6		r19.23v 2937	. . . . . . 7
7	5, 6	sylibr 212	. . . . . 6
8	7	r19.21bi 2826	. . . . 5
9		eleq1 2529	. . . . 5
10	8, 9	mpbidi 216	. . . 4
11	10	exlimdv 1724	. . 3
12	3, 11	mpd 15	. 2
13	4	biimpa 484	. 2
14		ralxfr2d.3	. 2
15	12, 13, 14	ralxfrd 4666	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  A.wral 2807  E.wrex 2808

This theorem is referenced by:  rexxfr2d  4669  ralrn  6034  ralima  6152  cnrest2  19787  cnprest2  19791  consuba  19921  subislly  19982  trfbas2  20344  trfil2  20388  flimrest  20484  fclsrest  20525  tsmssubm  20644  metucn  21092  extoimad  37981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111