Theorem opelopabf 3572

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 3570 uses bound-variable hypotheses in place of distinct variable conditions."

Hypotheses

Ref	Expression
opelopabf.x	|- (ps -> A.xps)
opelopabf.y	|- (ch -> A.ych)
opelopabf.1	|- A e. _V
opelopabf.2	|- B e. _V
opelopabf.3	|- (x = A -> (ph <-> ps))
opelopabf.4	|- (y = B -> (ps <-> ch))

Assertion

Ref	Expression
opelopabf	|- (<.A, B>. e. {<.x, y>. | ph} <-> ch)

Distinct variable groups:   x,y,A   y,B

Proof of Theorem opelopabf

Step	Hyp	Ref	Expression
1		ax-17 1317	. . . 4 |- (ph -> A.zph)
2		ax-17 1317	. . . 4 |- (ph -> A.wph)
3		hbs1 1722	. . . . 5 |- ([z / x]ph -> A.x[z / x]ph)
4	3	hbsb 1723	. . . 4 |- ([w / y][z / x]ph -> A.x[w / y][z / x]ph)
5		hbs1 1722	. . . 4 |- ([w / y][z / x]ph -> A.y[w / y][z / x]ph)
6		sbequ12 1545	. . . . 5 |- (x = z -> (ph <-> [z / x]ph))
7		sbequ12 1545	. . . . 5 |- (y = w -> ([z / x]ph <-> [w / y][z / x]ph))
8	6, 7	sylan9bb 599	. . . 4 |- ((x = z /\ y = w) -> (ph <-> [w / y][z / x]ph))
9	1, 2, 4, 5, 8	cbvopab 3403	. . 3 |- {<.x, y>. | ph} = {<.z, w>. | [w / y][z / x]ph}
10	9	eleq2i 1961	. 2 |- (<.A, B>. e. {<.x, y>. | ph} <-> <.A, B>. e. {<.z, w>. | [w / y][z / x]ph})
11		opelopabf.1	. . 3 |- A e. _V
12		opelopabf.2	. . 3 |- B e. _V
13		ax-17 1317	. . . 4 |- (z = A -> A.y z = A)
14		opelopabf.x	. . . . 5 |- (ps -> A.xps)
15		opelopabf.3	. . . . 5 |- (x = A -> (ph <-> ps))
16	14, 15	sbhypf 2452	. . . 4 |- (z = A -> ([z / x]ph <-> ps))
17	13, 16	sbbid 1617	. . 3 |- (z = A -> ([w / y][z / x]ph <-> [w / y]ps))
18		opelopabf.y	. . . 4 |- (ch -> A.ych)
19		opelopabf.4	. . . 4 |- (y = B -> (ps <-> ch))
20	18, 19	sbhypf 2452	. . 3 |- (w = B -> ([w / y]ps <-> ch))
21	11, 12, 17, 20	opelopab 3570	. 2 |- (<.A, B>. e. {<.z, w>. | [w / y][z / x]ph} <-> ch)
22	10, 21	bitri 190	1 |- (<.A, B>. e. {<.x, y>. | ph} <-> ch)

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534  _Vcvv 2292  <.cop 3046  {copab 3395

This theorem is referenced by:  iunfopabOLD 4543  pofun 15772

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-opab 3396

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