Theorem 3gencl 3141

Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Hypotheses

Ref	Expression
3gencl.1
3gencl.2
3gencl.3
3gencl.4
3gencl.5
3gencl.6
3gencl.7

Assertion

Ref	Expression
3gencl

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

Proof of Theorem 3gencl

Step	Hyp	Ref	Expression
1		3gencl.3	. . . . 5
2		df-rex 2813	. . . . 5
3	1, 2	bitri 249	. . . 4
4		3gencl.6	. . . . 5
5	4	imbi2d 316	. . . 4
6		3gencl.1	. . . . . 6
7		3gencl.2	. . . . . 6
8		3gencl.4	. . . . . . 7
9	8	imbi2d 316	. . . . . 6
10		3gencl.5	. . . . . . 7
11	10	imbi2d 316	. . . . . 6
12		3gencl.7	. . . . . . 7
13	12	3expia 1198	. . . . . 6
14	6, 7, 9, 11, 13	2gencl 3140	. . . . 5
15	14	com12 31	. . . 4
16	3, 5, 15	gencl 3139	. . 3
17	16	com12 31	. 2
18	17	3impia 1193	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  E.wex 1612  e.wcel 1818  E.wrex 2808

This theorem is referenced by:  axpre-lttrn  9564  axpre-ltadd  9565

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

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-ex 1613  df-rex 2813