Theorem hboprd 4905

Description: Deduction version of bound-variable hypothesis builder hbopr 4904. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)

Hypotheses

Ref	Expression
hboprd.1	|- (ph -> A.xph)
hboprd.2	|- (ph -> (y e. A -> A.x y e. A))
hboprd.3	|- (ph -> (y e. F -> A.x y e. F))
hboprd.4	|- (ph -> (y e. B -> A.x y e. B))

Assertion

Ref	Expression
hboprd	|- (ph -> (y e. (AFB) -> A.x y e. (AFB)))

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

Proof of Theorem hboprd

Step	Hyp	Ref	Expression
1		hboprd.1	. . 3 |- (ph -> A.xph)
2		hboprd.3	. . 3 |- (ph -> (y e. F -> A.x y e. F))
3		hboprd.2	. . . 4 |- (ph -> (y e. A -> A.x y e. A))
4		hboprd.4	. . . 4 |- (ph -> (y e. B -> A.x y e. B))
5	1, 3, 4	hbopd 3169	. . 3 |- (ph -> (y e. <.A, B>. -> A.x y e. <.A, B>.))
6	1, 2, 5	hbfvd 4687	. 2 |- (ph -> (y e. (F` <.A, B>.) -> A.x y e. (F` <.A, B>.)))
7		df-opr 4886	. . 3 |- (AFB) = (F` <.A, B>.)
8	7	eleq2i 1961	. 2 |- (y e. (AFB) <-> y e. (F` <.A, B>.))
9	8	albii 1346	. 2 |- (A.x y e. (AFB) <-> A.x y e. (F` <.A, B>.))
10	6, 8, 9	3imtr4g 612	1 |- (ph -> (y e. (AFB) -> A.x y e. (AFB)))

Syntax hints:   -> wi 3  A.wal 1296   e. wcel 1300  <.cop 3046  ` cfv 3998  (class class class)co 4884

This theorem is referenced by:  csboprgOLD 4911  hbnegd 6518

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-rex 2110  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-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886

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