Theorem hbcsb1g 2020

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 1813	. 2 |- (A e. C -> A e. V)
2		hbcsb1g.1	. . . . . 6 |- (y e. A -> A.x y e. A)
3		ax-17 969	. . . . . 6 |- (y e. V -> A.x y e. V)
4	2, 3	hbel 1563	. . . . 5 |- (A e. V -> A.x A e. V)
5		ax-17 969	. . . . 5 |- (A e. V -> A.z A e. V)
6	4, 5	19.21ai 996	. . . 4 |- (A e. V -> A.xA.z A e. V)
7	2	hbsbc1g 1944	. . . 4 |- (A e. V -> ([A / x]z e. B -> A.x[A / x]z e. B))
8	6, 7	hbabd 1466	. . 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 1998	. . . 4 |- [_A / x]_B = {z | [A / x]z e. B}
10	9	eleq2i 1535	. . 3 |- (y e. [_A / x]_B <-> y e. {z | [A / x]z e. B})
11	10	albii 997	. . 3 |- (A.x y e. [_A / x]_B <-> A.x y e. {z | [A / x]z e. B})
12	8, 10, 11	3imtr4g 552	. 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 952   e. wcel 956  [wsbc 1168  {cab 1461  Vcvv 1807  [_csb 1997

This theorem is referenced by:  hbcsb1 2021  csbnestglem 2031  csbnest1g 2033  sbcbrg 2657  csbima12g 3405  csbfv12g 3733  csboprg 3977  csbnegg 5344  fsum0diaglem2 7200  fsum0diag 7201

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-10 964  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-an 225  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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