Theorem 3gencl 1830

Description: Implicit substitution for class with embedded variable.

Hypotheses

Ref	Expression
3gencl.1	|- (D e. S <-> E.x(x e. R /\ A = D))
3gencl.2	|- (F e. S <-> E.y(y e. R /\ B = F))
3gencl.3	|- (G e. S <-> E.z(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

Proof of Theorem 3gencl

Step	Hyp	Ref	Expression
1		3gencl.3	. . . 4 |- (G e. S <-> E.z(z e. R /\ C = G))
2		3gencl.6	. . . . 5 |- (C = G -> (ch <-> th))
3	2	imbi2d 612	. . . 4 |- (C = G -> (((D e. S /\ F e. S) -> ch) <-> ((D e. S /\ F e. S) -> th)))
4		3gencl.1	. . . . . 6 |- (D e. S <-> E.x(x e. R /\ A = D))
5		3gencl.2	. . . . . 6 |- (F e. S <-> E.y(y e. R /\ B = F))
6		3gencl.4	. . . . . . 7 |- (A = D -> (ph <-> ps))
7	6	imbi2d 612	. . . . . 6 |- (A = D -> ((z e. R -> ph) <-> (z e. R -> ps)))
8		3gencl.5	. . . . . . 7 |- (B = F -> (ps <-> ch))
9	8	imbi2d 612	. . . . . 6 |- (B = F -> ((z e. R -> ps) <-> (z e. R -> ch)))
10		3gencl.7	. . . . . . 7 |- ((x e. R /\ y e. R /\ z e. R) -> ph)
11	10	3expia 835	. . . . . 6 |- ((x e. R /\ y e. R) -> (z e. R -> ph))
12	4, 5, 7, 9, 11	2gencl 1829	. . . . 5 |- ((D e. S /\ F e. S) -> (z e. R -> ch))
13	12	com12 11	. . . 4 |- (z e. R -> ((D e. S /\ F e. S) -> ch))
14	1, 3, 13	gencl 1828	. . 3 |- (G e. S -> ((D e. S /\ F e. S) -> th))
15	14	com12 11	. 2 |- ((D e. S /\ F e. S) -> (G e. S -> th))
16	15	3impia 830	1 |- ((D e. S /\ F e. S /\ G e. S) -> th)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wex 980

This theorem is referenced by:  ltsor 5261  pre-axltadd 5289

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ex 981

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