Theorem hbcsbgd 2024

Description: Deduction version of hbcsbg 2022.

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 12	. . . . 5 |- (ph -> (ph -> A.xph))
3		hbcsbgd.3	. . . . . 6 |- (ph -> (z e. A -> A.x z e. A))
4		ax-17 969	. . . . . . 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 1910	. . . . 5 |- (ph -> (A e. V -> A.x A e. V))
7	2, 6	hband 1109	. . . 4 |- (ph -> ((ph /\ A e. V) -> A.x(ph /\ A e. V)))
8	7	anabsi5 495	. . 3 |- ((ph /\ A e. V) -> A.x(ph /\ A e. V))
9		ax-17 969	. . . 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-17 969	. . . . . . 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 1910	. . . . 5 |- (ph -> (w e. B -> A.x w e. B))
16	1, 11, 3, 15	hbsbcgd 1980	. . . 4 |- ((ph /\ A e. V) -> ([A / y]w e. B -> A.x[A / y]w e. B))
17		sbcel2g 2011	. . . . 5 |- (A e. V -> ([A / y]w e. B <-> w e. [_A / y]_B))
18	17	adantl 388	. . . 4 |- ((ph /\ A e. V) -> ([A / y]w e. B <-> w e. [_A / y]_B))
19	8, 18	albid 1102	. . . 4 |- ((ph /\ A e. V) -> (A.x[A / y]w e. B <-> A.x w e. [_A / y]_B))
20	16, 18, 19	3imtr3d 541	. . 3 |- ((ph /\ A e. V) -> (w e. [_A / y]_B -> A.x w e. [_A / y]_B))
21	8, 10, 20	hbeld 1910	. 2 |- ((ph /\ A e. V) -> (z e. [_A / y]_B -> A.x z e. [_A / y]_B))
22		elisset 1813	. 2 |- (A e. C -> A e. V)
23	21, 22	sylan2 451	1 |- ((ph /\ A e. C) -> (z e. [_A / y]_B -> A.x z e. [_A / y]_B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   e. wcel 956  [wsbc 1168  Vcvv 1807  [_csb 1997

This theorem is referenced by:  csbnestglem 2031

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-9 963  ax-10 964  ax-11 965  ax-12 966  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938  df-csb 1998

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