Theorem hbcsbg 2026

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 1817	. 2 |- (A e. C -> A e. V)
2		hbcsbg.1	. . . . . 6 |- (z e. A -> A.x z e. A)
3		ax-17 971	. . . . . 6 |- (z e. V -> A.x z e. V)
4	2, 3	hbel 1566	. . . . 5 |- (A e. V -> A.x A e. V)
5		ax-17 971	. . . . 5 |- (A e. V -> A.w A e. V)
6	4, 5	19.21ai 998	. . . 4 |- (A e. V -> A.xA.w A e. V)
7		ax-17 971	. . . . . 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 1566	. . . . 5 |- (w e. B -> A.x w e. B)
10	2, 9	hbsbcg 1951	. . . 4 |- (A e. V -> ([A / y]w e. B -> A.x[A / y]w e. B))
11	6, 10	hbabd 1468	. . 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 2002	. . . 4 |- [_A / y]_B = {w | [A / y]w e. B}
13	12	eleq2i 1538	. . 3 |- (z e. [_A / y]_B <-> z e. {w | [A / y]w e. B})
14	13	albii 999	. . 3 |- (A.x z e. [_A / y]_B <-> A.x z e. {w | [A / y]w e. B})
15	11, 13, 14	3imtr4g 553	. 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 954   e. wcel 958  [wsbc 1170  {cab 1463  Vcvv 1811  [_csb 2001

This theorem is referenced by:  csbie2t 2033  oprabval2gf 4026  foprab2 4119

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-10 966  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-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002

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