Theorem hbcsbgd 2571

Description: Deduction version of hbcsbg 2569.

Hypotheses

Ref	Expression
hbcsbgd.1	|- (ph -> A.xph)
hbcsbgd.2	|- (ph -> A.yph)
hbcsbgd.3	|- (ph -> (z e. A -> A.x z e. A))
hbcsbgd.4	|- (ph -> (z e. B -> A.x z e. B))

Assertion

Ref	Expression
hbcsbgd	|- ((ph /\ A e. C) -> (z e. [_A / y]_B -> A.x z e. [_A / y]_B))

Distinct variable groups:   z,A   z,B   ph,z   x,z

Proof of Theorem hbcsbgd

Step	Hyp	Ref	Expression
1		hbcsbgd.1	. . . . . 6 |- (ph -> A.xph)
2	1	a1d 15	. . . . 5 |- (ph -> (ph -> A.xph))
3		hbcsbgd.3	. . . . . 6 |- (ph -> (z e. A -> A.x z e. A))
4		ax-17 1317	. . . . . . 7 |- (z e. _V -> A.x z e. _V)
5	4	a1i 8	. . . . . 6 |- (ph -> (z e. _V -> A.x z e. _V))
6	1, 3, 5	hbeld 2425	. . . . 5 |- (ph -> (A e. _V -> A.x A e. _V))
7	2, 6	hband 1469	. . . 4 |- (ph -> ((ph /\ A e. _V) -> A.x(ph /\ A e. _V)))
8	7	anabsi5 553	. . 3 |- ((ph /\ A e. _V) -> A.x(ph /\ A e. _V))
9		ax-17 1317	. . . 4 |- (w e. z -> A.x w e. z)
10	9	a1i 8	. . 3 |- ((ph /\ A e. _V) -> (w e. z -> A.x w e. z))
11		hbcsbgd.2	. . . . 5 |- (ph -> A.yph)
12		ax-17 1317	. . . . . . 7 |- (z e. w -> A.x z e. w)
13	12	a1i 8	. . . . . 6 |- (ph -> (z e. w -> A.x z e. w))
14		hbcsbgd.4	. . . . . 6 |- (ph -> (z e. B -> A.x z e. B))
15	1, 13, 14	hbeld 2425	. . . . 5 |- (ph -> (w e. B -> A.x w e. B))
16	1, 11, 3, 15	hbsbcgd 2517	. . . 4 |- ((ph /\ A e. _V) -> ([A / y]w e. B -> A.x[A / y]w e. B))
17		sbcel2g 2558	. . . . 5 |- (A e. _V -> ([A / y]w e. B <-> w e. [_A / y]_B))
18	17	adantl 424	. . . 4 |- ((ph /\ A e. _V) -> ([A / y]w e. B <-> w e. [_A / y]_B))
19	8, 18	albid 1459	. . . 4 |- ((ph /\ A e. _V) -> (A.x[A / y]w e. B <-> A.x w e. [_A / y]_B))
20	16, 18, 19	3imtr3d 601	. . 3 |- ((ph /\ A e. _V) -> (w e. [_A / y]_B -> A.x w e. [_A / y]_B))
21	8, 10, 20	hbeld 2425	. 2 |- ((ph /\ A e. _V) -> (z e. [_A / y]_B -> A.x z e. [_A / y]_B))
22		elisset 2299	. 2 |- (A e. C -> A e. _V)
23	21, 22	sylan2 500	1 |- ((ph /\ A e. C) -> (z e. [_A / y]_B -> A.x z e. [_A / y]_B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300  [wsbc 1534  _Vcvv 2292  [_csb 2540

This theorem is referenced by:  csbnestglem 2580

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454  df-csb 2541

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