Theorem opelopabg 2115

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61.

Hypotheses

Ref	Expression
opelopabg.1	|- (x = A -> (ph <-> ps))
opelopabg.2	|- (y = B -> (ps <-> ch))

Assertion

Ref	Expression
opelopabg	|- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | ph} <-> ch))

Distinct variable group(s):   x,y,A   x,B,y   ch,x,y

Proof of Theorem opelopabg

Step	Hyp	Ref	Expression
1		elex 1356	. . . 4 |- (A e. C -> E.x x = A)
2		elex 1356	. . . 4 |- (B e. D -> E.y y = B)
3	1, 2	anim12i 268	. . 3 |- ((A e. C /\ B e. D) -> (E.x x = A /\ E.y y = B))
4		eeanv 980	. . 3 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
5	3, 4	sylibr 175	. 2 |- ((A e. C /\ B e. D) -> E.xE.y(x = A /\ y = B))
6		ax-17 925	. . . . 5 |- (z e. <.A, B>. -> A.x z e. <.A, B>.)
7		hbopab1 2112	. . . . 5 |- (z e. {<.x, y>. | ph} -> A.x z e. {<.x, y>. | ph})
8	6, 7	hbel 1172	. . . 4 |- (<.A, B>. e. {<.x, y>. | ph} -> A.x<.A, B>. e. {<.x, y>. | ph})
9		ax-17 925	. . . 4 |- (ch -> A.xch)
10	8, 9	hbbi 705	. . 3 |- ((<.A, B>. e. {<.x, y>. | ph} <-> ch) -> A.x(<.A, B>. e. {<.x, y>. | ph} <-> ch))
11		ax-17 925	. . . . . 6 |- (z e. <.A, B>. -> A.y z e. <.A, B>.)
12		hbopab2 2113	. . . . . 6 |- (z e. {<.x, y>. | ph} -> A.y z e. {<.x, y>. | ph})
13	11, 12	hbel 1172	. . . . 5 |- (<.A, B>. e. {<.x, y>. | ph} -> A.y<.A, B>. e. {<.x, y>. | ph})
14		ax-17 925	. . . . 5 |- (ch -> A.ych)
15	13, 14	hbbi 705	. . . 4 |- ((<.A, B>. e. {<.x, y>. | ph} <-> ch) -> A.y(<.A, B>. e. {<.x, y>. | ph} <-> ch))
16		opeq12 1878	. . . . . . 7 |- ((x = A /\ y = B) -> <.x, y>. = <.A, B>.)
17	16	eleq1d 1155	. . . . . 6 |- ((x = A /\ y = B) -> (<.x, y>. e. {<.x, y>. | ph} <-> <.A, B>. e. {<.x, y>. | ph}))
18		opabid 2099	. . . . . 6 |- (<.x, y>. e. {<.x, y>. | ph} <-> ph)
19	17, 18	syl5bbr 412	. . . . 5 |- ((x = A /\ y = B) -> (ph <-> <.A, B>. e. {<.x, y>. | ph}))
20		opelopabg.1	. . . . . 6 |- (x = A -> (ph <-> ps))
21		opelopabg.2	. . . . . 6 |- (y = B -> (ps <-> ch))
22	20, 21	sylan9bb 418	. . . . 5 |- ((x = A /\ y = B) -> (ph <-> ch))
23	19, 22	bitr3d 408	. . . 4 |- ((x = A /\ y = B) -> (<.A, B>. e. {<.x, y>. | ph} <-> ch))
24	15, 23	19.23ai 746	. . 3 |- (E.y(x = A /\ y = B) -> (<.A, B>. e. {<.x, y>. | ph} <-> ch))
25	10, 24	19.23ai 746	. 2 |- (E.xE.y(x = A /\ y = B) -> (<.A, B>. e. {<.x, y>. | ph} <-> ch))
26	5, 25	syl 12	1 |- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | ph} <-> ch))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  <.cop 1810  {copab 2055

This theorem is referenced by:  brabg 2116  opelopab 2117  opelcnvg 2517  fvopab3 2868  fvopab3ig 2869  fvopabn 2873  oprabval 3047  brecop 3242

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

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