Theorem hbcsb1gd 2023

Description: Deduction version of hbcsb1g 2020.

Hypotheses

Ref	Expression
hbcsb1gd.1	|- (ph -> A.xph)
hbcsb1gd.2	|- (ph -> (y e. A -> A.x y e. A))

Assertion

Ref	Expression
hbcsb1gd	|- ((ph /\ A e. C) -> (y e. [_A / x]_B -> A.x y e. [_A / x]_B))

Distinct variable groups:   y,A   ph,y   x,y

Proof of Theorem hbcsb1gd

Step	Hyp	Ref	Expression
1		hbcsb1gd.1	. . . . . 6 |- (ph -> A.xph)
2	1	a1d 12	. . . . 5 |- (ph -> (ph -> A.xph))
3		hbcsb1gd.2	. . . . . 6 |- (ph -> (y e. A -> A.x y e. A))
4		ax-17 969	. . . . . . 7 |- (y e. V -> A.x y e. V)
5	4	a1i 8	. . . . . 6 |- (ph -> (y e. V -> A.x y 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 |- (z e. y -> A.x z e. y)
10	9	a1i 8	. . 3 |- ((ph /\ A e. V) -> (z e. y -> A.x z e. y))
11	1, 3	hbsbc1gd 1979	. . . 4 |- ((ph /\ A e. V) -> ([A / x]z e. B -> A.x[A / x]z e. B))
12		sbcel2g 2011	. . . . 5 |- (A e. V -> ([A / x]z e. B <-> z e. [_A / x]_B))
13	12	adantl 388	. . . 4 |- ((ph /\ A e. V) -> ([A / x]z e. B <-> z e. [_A / x]_B))
14	8, 13	albid 1102	. . . 4 |- ((ph /\ A e. V) -> (A.x[A / x]z e. B <-> A.x z e. [_A / x]_B))
15	11, 13, 14	3imtr3d 541	. . 3 |- ((ph /\ A e. V) -> (z e. [_A / x]_B -> A.x z e. [_A / x]_B))
16	8, 10, 15	hbeld 1910	. 2 |- ((ph /\ A e. V) -> (y e. [_A / x]_B -> A.x y e. [_A / x]_B))
17		elisset 1813	. 2 |- (A e. C -> A e. V)
18	16, 17	sylan2 451	1 |- ((ph /\ A e. C) -> (y e. [_A / x]_B -> A.x y e. [_A / x]_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:  csbnest1g 2033

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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