Theorem hbfvd2 3722

Description: Deduction version of bound-variable hypothesis builder hbfv 3720. This variant of hbfvd 3721 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 1001	. . . . 5 |- (A.x z e. F -> A.xA.x z e. F)
2	1	hbab 1465	. . . 4 |- (y e. {z | A.x z e. F} -> A.x y e. {z | A.x z e. F})
3		hba1 1001	. . . . 5 |- (A.x z e. A -> A.xA.x z e. A)
4	3	hbab 1465	. . . 4 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
5	2, 4	hbfv 3720	. . 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 1058	. . . . . . 7 |- (ph -> A.yph)
9		hbfvd2.2	. . . . . . 7 |- (ph -> (y e. F -> A.x y e. F))
10	8, 9	19.21ai 996	. . . . . 6 |- (ph -> A.y(y e. F -> A.x y e. F))
11		abidhb 1908	. . . . . 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 3717	. . . 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 996	. . . . . 6 |- (ph -> A.y(y e. A -> A.x y e. A))
16		abidhb 1908	. . . . . 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 3719	. . . 4 |- (ph -> (F` {z | A.x z e. A}) = (F` A))
19	13, 18	eqtrd 1504	. . 3 |- (ph -> ({z | A.x z e. F}` {z | A.x z e. A}) = (F` A))
20	19	eleq2d 1538	. 2 |- (ph -> (y e. ({z | A.x z e. F}` {z | A.x z e. A}) <-> y e. (F` A)))
21		ax-4 971	. . . . 5 |- (A.yA.xph -> A.xph)
22	21	a7s 989	. . . 4 |- (A.xA.yph -> A.xph)
23	7, 22	syl 10	. . 3 |- (ph -> A.xph)
24	23, 20	albid 1102	. 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 541	1 |- (ph -> (y e. (F` A) -> A.x y e. (F` A)))

Syntax hints:   -> wi 3  A.wal 952   = wceq 954   e. wcel 956  {cab 1461  ` cfv 3177

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193

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