Theorem opabsb 2815

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 groups:   x,y,z   x,w,y

Proof of Theorem opabsb

Step	Hyp	Ref	Expression
1		a9e 1125	. 2 |- E.y y = w
2		ax-17 971	. . . . 5 |- (v e. <.z, w>. -> A.y v e. <.z, w>.)
3		hbopab2 2814	. . . . 5 |- (v e. {<.x, y>. | ph} -> A.y v e. {<.x, y>. | ph})
4	2, 3	hbel 1566	. . . 4 |- (<.z, w>. e. {<.x, y>. | ph} -> A.y<.z, w>. e. {<.x, y>. | ph})
5		hbs1 1332	. . . 4 |- ([w / y][z / x]ph -> A.y[w / y][z / x]ph)
6	4, 5	hbbi 1010	. . 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 1125	. . . 4 |- E.x x = z
8		ax-17 971	. . . . . 6 |- (y = w -> A.x y = w)
9		ax-17 971	. . . . . . . 8 |- (v e. <.z, w>. -> A.x v e. <.z, w>.)
10		hbopab1 2813	. . . . . . . 8 |- (v e. {<.x, y>. | ph} -> A.x v e. {<.x, y>. | ph})
11	9, 10	hbel 1566	. . . . . . 7 |- (<.z, w>. e. {<.x, y>. | ph} -> A.x<.z, w>. e. {<.x, y>. | ph})
12		hbs1 1332	. . . . . . . 8 |- ([z / x]ph -> A.x[z / x]ph)
13	12	hbsb 1333	. . . . . . 7 |- ([w / y][z / x]ph -> A.x[w / y][z / x]ph)
14	11, 13	hbbi 1010	. . . . . 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 1007	. . . . 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 2489	. . . . . . . . 9 |- ((x = z /\ y = w) -> <.x, y>. = <.z, w>.)
17	16	eleq1d 1540	. . . . . . . 8 |- ((x = z /\ y = w) -> (<.x, y>. e. {<.x, y>. | ph} <-> <.z, w>. e. {<.x, y>. | ph}))
18		opabid 2810	. . . . . . . 8 |- (<.x, y>. e. {<.x, y>. | ph} <-> ph)
19	17, 18	syl5bbr 534	. . . . . . 7 |- ((x = z /\ y = w) -> (ph <-> <.z, w>. e. {<.x, y>. | ph}))
20		sbequ12 1181	. . . . . . . 8 |- (x = z -> (ph <-> [z / x]ph))
21		sbequ12 1181	. . . . . . . 8 |- (y = w -> ([z / x]ph <-> [w / y][z / x]ph))
22	20, 21	sylan9bb 540	. . . . . . 7 |- ((x = z /\ y = w) -> (ph <-> [w / y][z / x]ph))
23	19, 22	bitr3d 530	. . . . . 6 |- ((x = z /\ y = w) -> (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph))
24	23	ex 373	. . . . 5 |- (x = z -> (y = w -> (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph)))
25	15, 24	19.23ai 1064	. . . 4 |- (E.x x = z -> (y = w -> (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph)))
26	7, 25	ax-mp 7	. . 3 |- (y = w -> (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph))
27	6, 26	19.23ai 1064	. 2 |- (E.y y = w -> (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph))
28	1, 27	ax-mp 7	1 |- (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  [wsbc 1170  <.cop 2411  {copab 2666

This theorem is referenced by:  brabsb 2816  inopab 3268  isarep1 3577

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667

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