Theorem opabsb 2114

Description: The law of concretion in terms of substitutions.

Assertion

Ref	Expression
opabsb	|- (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph)

Distinct variable group(s):   x,y,z,w

Proof of Theorem opabsb

Step	Hyp	Ref	Expression
1		a9e 809	. 2 |- E.y y = w
2		ax-17 925	. . . . 5 |- (v e. <.z, w>. -> A.y v e. <.z, w>.)
3		hbopab2 2113	. . . . 5 |- (v e. {<.x, y>. | ph} -> A.y v e. {<.x, y>. | ph})
4	2, 3	hbel 1172	. . . 4 |- (<.z, w>. e. {<.x, y>. | ph} -> A.y<.z, w>. e. {<.x, y>. | ph})
5		hbs1 986	. . . 4 |- ([w / y][z / x]ph -> A.y[w / y][z / x]ph)
6	4, 5	hbbi 705	. . 3 |- ((<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph) -> A.y(<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph))
7		a9e 809	. . . 4 |- E.x x = z
8		ax-17 925	. . . . . 6 |- (y = w -> A.x y = w)
9		ax-17 925	. . . . . . . 8 |- (v e. <.z, w>. -> A.x v e. <.z, w>.)
10		hbopab1 2112	. . . . . . . 8 |- (v e. {<.x, y>. | ph} -> A.x v e. {<.x, y>. | ph})
11	9, 10	hbel 1172	. . . . . . 7 |- (<.z, w>. e. {<.x, y>. | ph} -> A.x<.z, w>. e. {<.x, y>. | ph})
12		hbs1 986	. . . . . . . 8 |- ([z / x]ph -> A.x[z / x]ph)
13	12	hbsb 987	. . . . . . 7 |- ([w / y][z / x]ph -> A.x[w / y][z / x]ph)
14	11, 13	hbbi 705	. . . . . 6 |- ((<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph) -> A.x(<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph))
15	8, 14	hbim 702	. . . . 5 |- ((y = w -> (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph)) -> A.x(y = w -> (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph)))
16		opeq12 1878	. . . . . . . . 9 |- ((x = z /\ y = w) -> <.x, y>. = <.z, w>.)
17	16	eleq1d 1155	. . . . . . . 8 |- ((x = z /\ y = w) -> (<.x, y>. e. {<.x, y>. | ph} <-> <.z, w>. e. {<.x, y>. | ph}))
18		opabid 2099	. . . . . . . 8 |- (<.x, y>. e. {<.x, y>. | ph} <-> ph)
19	17, 18	syl5bbr 412	. . . . . . 7 |- ((x = z /\ y = w) -> (ph <-> <.z, w>. e. {<.x, y>. | ph}))
20		sbequ12 865	. . . . . . . 8 |- (x = z -> (ph <-> [z / x]ph))
21		sbequ12 865	. . . . . . . 8 |- (y = w -> ([z / x]ph <-> [w / y][z / x]ph))
22	20, 21	sylan9bb 418	. . . . . . 7 |- ((x = z /\ y = w) -> (ph <-> [w / y][z / x]ph))
23	19, 22	bitr3d 408	. . . . . 6 |- ((x = z /\ y = w) -> (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph))
24	23	exp 291	. . . . 5 |- (x = z -> (y = w -> (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph)))
25	15, 24	19.23ai 746	. . . 4 |- (E.x x = z -> (y = w -> (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph)))
26	7, 25	ax-mp 6	. . 3 |- (y = w -> (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph))
27	6, 26	19.23ai 746	. 2 |- (E.y y = w -> (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph))
28	1, 27	ax-mp 6	1 |- (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = weq 797  [wsb 852   e. wcel 1092  <.cop 1810  {copab 2055

This theorem is referenced by:  inopab 2495

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

metamath.org