Theorem hbfvd2 4688

Description: Deduction version of bound-variable hypothesis builder hbfv 4686. This variant of hbfvd 4687 allows us to create a closed theorem form by replacing the uncommitted antecedent ph with an appropriate substitution instance.

Hypotheses

Ref	Expression
hbfvd2.1	|- (ph -> A.xA.yph)
hbfvd2.2	|- (ph -> (y e. F -> A.x y e. F))
hbfvd2.3	|- (ph -> (y e. A -> A.x y e. A))

Assertion

Ref	Expression
hbfvd2	|- (ph -> (y e. (F` A) -> A.x y e. (F` A)))

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

Proof of Theorem hbfvd2

Step	Hyp	Ref	Expression
1		hba1 1350	. . . . 5 |- (A.x z e. F -> A.xA.x z e. F)
2	1	hbab 1875	. . . 4 |- (y e. {z | A.x z e. F} -> A.x y e. {z | A.x z e. F})
3		hba1 1350	. . . . 5 |- (A.x z e. A -> A.xA.x z e. A)
4	3	hbab 1875	. . . 4 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
5	2, 4	hbfv 4686	. . 3 |- (y e. ({z | A.x z e. F}` {z | A.x z e. A}) -> A.x y e. ({z | A.x z e. F}` {z | A.x z e. A}))
6	5	a1i 8	. 2 |- (ph -> (y e. ({z | A.x z e. F}` {z | A.x z e. A}) -> A.x y e. ({z | A.x z e. F}` {z | A.x z e. A})))
7		hbfvd2.1	. . . . . . . 8 |- (ph -> A.xA.yph)
8	7	19.21bi 1408	. . . . . . 7 |- (ph -> A.yph)
9		hbfvd2.2	. . . . . . 7 |- (ph -> (y e. F -> A.x y e. F))
10	8, 9	19.21ai 1345	. . . . . 6 |- (ph -> A.y(y e. F -> A.x y e. F))
11		abidhb 2423	. . . . . 6 |- (A.y(y e. F -> A.x y e. F) -> {z | A.x z e. F} = F)
12	10, 11	syl 12	. . . . 5 |- (ph -> {z | A.x z e. F} = F)
13	12	fveq1d 4683	. . . 4 |- (ph -> ({z | A.x z e. F}` {z | A.x z e. A}) = (F` {z | A.x z e. A}))
14		hbfvd2.3	. . . . . . 7 |- (ph -> (y e. A -> A.x y e. A))
15	8, 14	19.21ai 1345	. . . . . 6 |- (ph -> A.y(y e. A -> A.x y e. A))
16		abidhb 2423	. . . . . 6 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
17	15, 16	syl 12	. . . . 5 |- (ph -> {z | A.x z e. A} = A)
18	17	fveq2d 4685	. . . 4 |- (ph -> (F` {z | A.x z e. A}) = (F` A))
19	13, 18	eqtrd 1925	. . 3 |- (ph -> ({z | A.x z e. F}` {z | A.x z e. A}) = (F` A))
20	19	eleq2d 1964	. 2 |- (ph -> (y e. ({z | A.x z e. F}` {z | A.x z e. A}) <-> y e. (F` A)))
21		ax-4 1319	. . . . 5 |- (A.yA.xph -> A.xph)
22	21	a7s 1337	. . . 4 |- (A.xA.yph -> A.xph)
23	7, 22	syl 12	. . 3 |- (ph -> A.xph)
24	23, 20	albid 1459	. 2 |- (ph -> (A.x y e. ({z | A.x z e. F}` {z | A.x z e. A}) <-> A.x y e. (F` A)))
25	6, 20, 24	3imtr3d 601	1 |- (ph -> (y e. (F` A) -> A.x y e. (F` A)))

Syntax hints:   -> wi 3  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871  ` cfv 3998

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

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