Theorem 2optocl 2470

Description: Implicit substitution of classes for ordered pairs.

Hypotheses

Ref	Expression
2optocl.1	|- R = (C X. D)
2optocl.2	|- (<.x, y>. = A -> (ph <-> ps))
2optocl.3	|- (<.z, w>. = B -> (ps <-> ch))
2optocl.4	|- (((x e. C /\ y e. D) /\ (z e. C /\ w e. D)) -> ph)

Assertion

Ref	Expression
2optocl	|- ((A e. R /\ B e. R) -> ch)

Distinct variable group(s):   x,y,z,w,A   z,B,w   x,C,y,z,w   x,D,y,z,w   ps,x,y   ch,z,w   z,R,w

Proof of Theorem 2optocl

Step	Hyp	Ref	Expression
1		2optocl.1	. . . 4 |- R = (C X. D)
2		2optocl.3	. . . . 5 |- (<.z, w>. = B -> (ps <-> ch))
3	2	imbi2d 464	. . . 4 |- (<.z, w>. = B -> ((A e. R -> ps) <-> (A e. R -> ch)))
4		2optocl.2	. . . . . . 7 |- (<.x, y>. = A -> (ph <-> ps))
5	4	imbi2d 464	. . . . . 6 |- (<.x, y>. = A -> (((z e. C /\ w e. D) -> ph) <-> ((z e. C /\ w e. D) -> ps)))
6		2optocl.4	. . . . . . 7 |- (((x e. C /\ y e. D) /\ (z e. C /\ w e. D)) -> ph)
7	6	exp 291	. . . . . 6 |- ((x e. C /\ y e. D) -> ((z e. C /\ w e. D) -> ph))
8	1, 5, 7	optocl 2469	. . . . 5 |- (A e. R -> ((z e. C /\ w e. D) -> ps))
9	8	com12 13	. . . 4 |- ((z e. C /\ w e. D) -> (A e. R -> ps))
10	1, 3, 9	optocl 2469	. . 3 |- (B e. R -> (A e. R -> ch))
11	10	com12 13	. 2 |- (A e. R -> (B e. R -> ch))
12	11	imp 277	1 |- ((A e. R /\ B e. R) -> ch)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  <.cop 1810   X. cxp 2408

This theorem is referenced by:  3optocl 2471  ecopoprsym 3246  th3qlem2 3251  axaddcl 4066  axmulcl 4068

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424

metamath.org