Theorem 3gencl 2890

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

Hypotheses

Ref	Expression
3gencl.1	|- (D e. S <-> E.x e. R A = D)
3gencl.2	|- (F e. S <-> E.y e. R B = F)
3gencl.3	|- (G e. S <-> E.z e. R C = G)
3gencl.4	|- (A = D -> (ph <-> ps))
3gencl.5	|- (B = F -> (ps <-> ch))
3gencl.6	|- (C = G -> (ch <-> th))
3gencl.7	|- ((x e. R /\ y e. R /\ z e. R) -> ph)

Assertion

Ref	Expression
3gencl	|- ((D e. S /\ F e. S /\ G e. S) -> th)

Distinct variable groups:   x,y,z   y,D,z   z,F   x,R,y   y,S,z   ps,x   ch,y   th,z

Allowed substitution hints:   ph(x,y,z)   ps(y,z)   ch(x,z)   th(x,y)   A(x,y,z)   B(x,y,z)   C(x,y,z)   D(x)   R(z)   S(x)   F(x,y)   G(x,y,z)

Proof of Theorem 3gencl

Step	Hyp	Ref	Expression
1		3gencl.3	. . . . 5 |- (G e. S <-> E.z e. R C = G)
2		df-rex 2621	. . . . 5 |- (E.z e. R C = G <-> E.z(z e. R /\ C = G))
3	1, 2	bitri 240	. . . 4 |- (G e. S <-> E.z(z e. R /\ C = G))
4		3gencl.6	. . . . 5 |- (C = G -> (ch <-> th))
5	4	imbi2d 307	. . . 4 |- (C = G -> (((D e. S /\ F e. S) -> ch) <-> ((D e. S /\ F e. S) -> th)))
6		3gencl.1	. . . . . 6 |- (D e. S <-> E.x e. R A = D)
7		3gencl.2	. . . . . 6 |- (F e. S <-> E.y e. R B = F)
8		3gencl.4	. . . . . . 7 |- (A = D -> (ph <-> ps))
9	8	imbi2d 307	. . . . . 6 |- (A = D -> ((z e. R -> ph) <-> (z e. R -> ps)))
10		3gencl.5	. . . . . . 7 |- (B = F -> (ps <-> ch))
11	10	imbi2d 307	. . . . . 6 |- (B = F -> ((z e. R -> ps) <-> (z e. R -> ch)))
12		3gencl.7	. . . . . . 7 |- ((x e. R /\ y e. R /\ z e. R) -> ph)
13	12	3expia 1153	. . . . . 6 |- ((x e. R /\ y e. R) -> (z e. R -> ph))
14	6, 7, 9, 11, 13	2gencl 2889	. . . . 5 |- ((D e. S /\ F e. S) -> (z e. R -> ch))
15	14	com12 27	. . . 4 |- (z e. R -> ((D e. S /\ F e. S) -> ch))
16	3, 5, 15	gencl 2888	. . 3 |- (G e. S -> ((D e. S /\ F e. S) -> th))
17	16	com12 27	. 2 |- ((D e. S /\ F e. S) -> (G e. S -> th))
18	17	3impia 1148	1 |- ((D e. S /\ F e. S /\ G e. S) -> th)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-rex 2621

This theorem is referenced by: (None)