Theorem ralxpxfr2d 3224

Description: Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Hypotheses

Ref	Expression
ralxpxfr2d.a
ralxpxfr2d.b
ralxpxfr2d.c

Assertion

Ref	Expression
ralxpxfr2d

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

Proof of Theorem ralxpxfr2d

Step	Hyp	Ref	Expression
1		df-ral 2812	. . . 4
2		ralxpxfr2d.b	. . . . . 6
3	2	imbi1d 317	. . . . 5
4	3	albidv 1713	. . . 4
5	1, 4	syl5bb 257	. . 3
6		ralcom4 3128	. . . 4
7		ralcom4 3128	. . . . 5
8	7	ralbii 2888	. . . 4
9		r19.23v 2937	. . . . . . 7
10	9	ralbii 2888	. . . . . 6
11		r19.23v 2937	. . . . . 6
12	10, 11	bitr2i 250	. . . . 5
13	12	albii 1640	. . . 4
14	6, 8, 13	3bitr4ri 278	. . 3
15	5, 14	syl6bb 261	. 2
16		ralxpxfr2d.c	. . . . . 6
17	16	pm5.74da 687	. . . . 5
18	17	albidv 1713	. . . 4
19		ralxpxfr2d.a	. . . . 5
20		biidd 237	. . . . 5
21	19, 20	ceqsalv 3137	. . . 4
22	18, 21	syl6bb 261	. . 3
23	22	2ralbidv 2901	. 2
24	15, 23	bitrd 253	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808  cvv 3109

This theorem is referenced by:  ralxpmap  7488

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-or 370  df-an 371  df-3an 975  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