Theorem hbcsb1g 2024

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

Hypothesis

Ref	Expression
hbcsb1g.1	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
hbcsb1g	|- (A e. C -> (y e. [_A / x]_B -> A.x y e. [_A / x]_B))

Distinct variable groups:   y,A   x,y

Proof of Theorem hbcsb1g

Step	Hyp	Ref	Expression
1		elisset 1817	. 2 |- (A e. C -> A e. V)
2		hbcsb1g.1	. . . . . 6 |- (y e. A -> A.x y e. A)
3		ax-17 971	. . . . . 6 |- (y e. V -> A.x y 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.z A e. V)
6	4, 5	19.21ai 998	. . . 4 |- (A e. V -> A.xA.z A e. V)
7	2	hbsbc1g 1948	. . . 4 |- (A e. V -> ([A / x]z e. B -> A.x[A / x]z e. B))
8	6, 7	hbabd 1468	. . 3 |- (A e. V -> (y e. {z | [A / x]z e. B} -> A.x y e. {z | [A / x]z e. B}))
9		df-csb 2002	. . . 4 |- [_A / x]_B = {z | [A / x]z e. B}
10	9	eleq2i 1538	. . 3 |- (y e. [_A / x]_B <-> y e. {z | [A / x]z e. B})
11	10	albii 999	. . 3 |- (A.x y e. [_A / x]_B <-> A.x y e. {z | [A / x]z e. B})
12	8, 10, 11	3imtr4g 553	. 2 |- (A e. V -> (y e. [_A / x]_B -> A.x y e. [_A / x]_B))
13	1, 12	syl 10	1 |- (A e. C -> (y e. [_A / x]_B -> A.x y e. [_A / x]_B))

Syntax hints:   -> wi 3  A.wal 954   e. wcel 958  [wsbc 1170  {cab 1463  Vcvv 1811  [_csb 2001

This theorem is referenced by:  hbcsb1 2025  csbnestglem 2035  csbnest1g 2037  sbcbrg 2662  csbima12g 3413  csbfv12g 3742  csboprg 3986  csbnegg 5364  fsum0diaglem2 7257  fsum0diag 7258

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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