Theorem hbcsb1 2025

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

Hypotheses

Ref	Expression
hbcsb1.1	|- A e. V
hbcsb1.2	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
hbcsb1	|- (y e. [_A / x]_B -> A.x y e. [_A / x]_B)

Distinct variable groups:   y,A   x,y

Proof of Theorem hbcsb1

Step	Hyp	Ref	Expression
1		hbcsb1.1	. 2 |- A e. V
2		hbcsb1.2	. . 3 |- (y e. A -> A.x y e. A)
3	2	hbcsb1g 2024	. 2 |- (A e. V -> (y e. [_A / x]_B -> A.x y e. [_A / x]_B))
4	1, 3	ax-mp 7	1 |- (y e. [_A / x]_B -> A.x y e. [_A / x]_B)

Syntax hints:   -> wi 3  A.wal 954   e. wcel 958  Vcvv 1811  [_csb 2001

This theorem is referenced by:  csbieb 2030  csbie2t 2033  uniiunlem 2132  sbcbrg 2662  csbima12g 3413  csbfv12g 3742  fvopab4gf 3781  fvopab4sf 3782  fvopabs 3792  fopabcos 3833  csboprg 3986  oprabval2gf 4026  foprab2 4119  csbnegg 5364  fsum1slem 7008  fsump1slem 7012  isumnn0nna 7208  infcvgaux1 7219  fsum0diaglem2 7257  fsum0diag2 7259  fsum0diag4 7261  iscaunns 7944

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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