Theorem brabg2 15681

Description: Relation by a binary relation abstraction.

Hypotheses

Ref	Expression
brabg2.1	|- (x = A -> (ph <-> ps))
brabg2.2	|- (y = B -> (ps <-> ch))
brabg2.3	|- R = {<.x, y>. | ph}
brabg2.4	|- (ch -> A e. C)

Assertion

Ref	Expression
brabg2	|- (B e. D -> (ARB <-> ch))

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

Proof of Theorem brabg2

Step	Hyp	Ref	Expression
1		relopab 4104	. . . . 5 |- Rel {<.x, y>. | ph}
2		brabg2.3	. . . . . 6 |- R = {<.x, y>. | ph}
3	2	releqi 4072	. . . . 5 |- (Rel R <-> Rel {<.x, y>. | ph})
4	1, 3	mpbir 207	. . . 4 |- Rel R
5	4	brrelexi 4029	. . 3 |- (ARB -> A e. _V)
6		brabg2.1	. . . . . . 7 |- (x = A -> (ph <-> ps))
7		brabg2.2	. . . . . . 7 |- (y = B -> (ps <-> ch))
8	6, 7, 2	brabg 3568	. . . . . 6 |- ((A e. _V /\ B e. D) -> (ARB <-> ch))
9	8	biimpd 170	. . . . 5 |- ((A e. _V /\ B e. D) -> (ARB -> ch))
10	9	ex 402	. . . 4 |- (A e. _V -> (B e. D -> (ARB -> ch)))
11	10	com3l 38	. . 3 |- (B e. D -> (ARB -> (A e. _V -> ch)))
12	5, 11	mpdi 59	. 2 |- (B e. D -> (ARB -> ch))
13		brabg2.4	. . 3 |- (ch -> A e. C)
14	6, 7, 2	brabg 3568	. . . . 5 |- ((A e. C /\ B e. D) -> (ARB <-> ch))
15	14	exbiri 421	. . . 4 |- (A e. C -> (B e. D -> (ch -> ARB)))
16	15	com3l 38	. . 3 |- (B e. D -> (ch -> (A e. C -> ARB)))
17	13, 16	mpdi 59	. 2 |- (B e. D -> (ch -> ARB))
18	12, 17	impbid 574	1 |- (B e. D -> (ARB <-> ch))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   class class class wbr 3338  {copab 3395  Rel wrel 3991

This theorem is referenced by:  tlmbr 15904

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-br 3339  df-opab 3396  df-xp 4000  df-rel 4001

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