Theorem hbcsbgd 2028

Description: Deduction version of hbcsbg 2026.

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-17 971	. . . . . . 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 1914	. . . . 5 |- (ph -> (A e. V -> A.x A e. V))
7	2, 6	hband 1111	. . . 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-17 971	. . . 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-17 971	. . . . . . 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 1914	. . . . 5 |- (ph -> (w e. B -> A.x w e. B))
16	1, 11, 3, 15	hbsbcgd 1984	. . . 4 |- ((ph /\ A e. V) -> ([A / y]w e. B -> A.x[A / y]w e. B))
17		sbcel2g 2015	. . . . 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 1104	. . . 4 |- ((ph /\ A e. V) -> (A.x[A / y]w e. B <-> A.x w e. [_A / y]_B))
20	16, 18, 19	3imtr3d 542	. . 3 |- ((ph /\ A e. V) -> (w e. [_A / y]_B -> A.x w e. [_A / y]_B))
21	8, 10, 20	hbeld 1914	. 2 |- ((ph /\ A e. V) -> (z e. [_A / y]_B -> A.x z e. [_A / y]_B))
22		elisset 1817	. 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 954   e. wcel 958  [wsbc 1170  Vcvv 1811  [_csb 2001

This theorem is referenced by:  csbnestglem 2035

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-9 965  ax-10 966  ax-11 967  ax-12 968  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002

Copyright terms: Public domain