Theorem hbcsbg 2016

Description: Bound-variable hypothesis builder for substitution into a class.

Hypotheses

Ref	Expression
hbcsbg.1	|- (z e. A -> A.x z e. A)
hbcsbg.2	|- (z e. B -> A.x z e. B)

Assertion

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

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

Proof of Theorem hbcsbg

Step	Hyp	Ref	Expression
1		elisset 1808	. 2 |- (A e. C -> A e. V)
2		hbcsbg.1	. . . . . 6 |- (z e. A -> A.x z e. A)
3		ax-17 968	. . . . . 6 |- (z e. V -> A.x z e. V)
4	2, 3	hbel 1558	. . . . 5 |- (A e. V -> A.x A e. V)
5		ax-17 968	. . . . 5 |- (A e. V -> A.w A e. V)
6	4, 5	19.21ai 995	. . . 4 |- (A e. V -> A.xA.w A e. V)
7		ax-17 968	. . . . . 6 |- (z e. w -> A.x z e. w)
8		hbcsbg.2	. . . . . 6 |- (z e. B -> A.x z e. B)
9	7, 8	hbel 1558	. . . . 5 |- (w e. B -> A.x w e. B)
10	2, 9	hbsbcg 1941	. . . 4 |- (A e. V -> ([A / y]w e. B -> A.x[A / y]w e. B))
11	6, 10	hbabd 1461	. . 3 |- (A e. V -> (z e. {w | [A / y]w e. B} -> A.x z e. {w | [A / y]w e. B}))
12		df-csb 1992	. . . 4 |- [_A / y]_B = {w | [A / y]w e. B}
13	12	eleq2i 1530	. . 3 |- (z e. [_A / y]_B <-> z e. {w | [A / y]w e. B})
14	13	albii 996	. . 3 |- (A.x z e. [_A / y]_B <-> A.x z e. {w | [A / y]w e. B})
15	11, 13, 14	3imtr4g 551	. 2 |- (A e. V -> (z e. [_A / y]_B -> A.x z e. [_A / y]_B))
16	1, 15	syl 10	1 |- (A e. C -> (z e. [_A / y]_B -> A.x z e. [_A / y]_B))

Syntax hints:   -> wi 3  A.wal 951   e. wcel 955  [wsbc 1166  {cab 1456  Vcvv 1802  [_csb 1991

This theorem is referenced by:  csbie2t 2023  oprabval2gf 4011  foprab2 4103

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932  df-csb 1992

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