Theorem hbfvd2 3731

Description: Deduction version of bound-variable hypothesis builder hbfv 3729. This variant of hbfvd 3730 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 1003	. . . . 5 |- (A.x z e. F -> A.xA.x z e. F)
2	1	hbab 1467	. . . 4 |- (y e. {z | A.x z e. F} -> A.x y e. {z | A.x z e. F})
3		hba1 1003	. . . . 5 |- (A.x z e. A -> A.xA.x z e. A)
4	3	hbab 1467	. . . 4 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
5	2, 4	hbfv 3729	. . 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 1060	. . . . . . 7 |- (ph -> A.yph)
9		hbfvd2.2	. . . . . . 7 |- (ph -> (y e. F -> A.x y e. F))
10	8, 9	19.21ai 998	. . . . . 6 |- (ph -> A.y(y e. F -> A.x y e. F))
11		abidhb 1912	. . . . . 6 |- (A.y(y e. F -> A.x y e. F) -> {z | A.x z e. F} = F)
12	10, 11	syl 10	. . . . 5 |- (ph -> {z | A.x z e. F} = F)
13	12	fveq1d 3726	. . . 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 998	. . . . . 6 |- (ph -> A.y(y e. A -> A.x y e. A))
16		abidhb 1912	. . . . . 6 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
17	15, 16	syl 10	. . . . 5 |- (ph -> {z | A.x z e. A} = A)
18	17	fveq2d 3728	. . . 4 |- (ph -> (F` {z | A.x z e. A}) = (F` A))
19	13, 18	eqtrd 1507	. . 3 |- (ph -> ({z | A.x z e. F}` {z | A.x z e. A}) = (F` A))
20	19	eleq2d 1541	. 2 |- (ph -> (y e. ({z | A.x z e. F}` {z | A.x z e. A}) <-> y e. (F` A)))
21		ax-4 973	. . . . 5 |- (A.yA.xph -> A.xph)
22	21	a7s 991	. . . 4 |- (A.xA.yph -> A.xph)
23	7, 22	syl 10	. . 3 |- (ph -> A.xph)
24	23, 20	albid 1104	. 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 542	1 |- (ph -> (y e. (F` A) -> A.x y e. (F` A)))

Syntax hints:   -> wi 3  A.wal 954   = wceq 956   e. wcel 958  {cab 1463  ` cfv 3182

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-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-rex 1650  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-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198

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