Theorem hbcsb1g 2567

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 2299	. 2 |- (A e. C -> A e. _V)
2		hbcsb1g.1	. . . . . 6 |- (y e. A -> A.x y e. A)
3		ax-17 1317	. . . . . 6 |- (y e. _V -> A.x y e. _V)
4	2, 3	hbel 1996	. . . . 5 |- (A e. _V -> A.x A e. _V)
5		ax-17 1317	. . . . 5 |- (A e. _V -> A.z A e. _V)
6	4, 5	19.21ai 1345	. . . 4 |- (A e. _V -> A.xA.z A e. _V)
7	2	hbsbc1g 2461	. . . 4 |- (A e. _V -> ([A / x]z e. B -> A.x[A / x]z e. B))
8	6, 7	hbabd 1876	. . 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 2541	. . . 4 |- [_A / x]_B = {z | [A / x]z e. B}
10	9	eleq2i 1961	. . 3 |- (y e. [_A / x]_B <-> y e. {z | [A / x]z e. B})
11	10	albii 1346	. . 3 |- (A.x y e. [_A / x]_B <-> A.x y e. {z | [A / x]z e. B})
12	8, 10, 11	3imtr4g 612	. 2 |- (A e. _V -> (y e. [_A / x]_B -> A.x y e. [_A / x]_B))
13	1, 12	syl 12	1 |- (A e. C -> (y e. [_A / x]_B -> A.x y e. [_A / x]_B))

Syntax hints:   -> wi 3  A.wal 1296   e. wcel 1300  [wsbc 1534  {cab 1871  _Vcvv 2292  [_csb 2540

This theorem is referenced by:  hbcsb1 2568  csbnestglem 2580  csbnest1g 2582  sbcbrgOLD 3391  csbima12g 4276  csbfv12g 4699  csboprgOLD 4911  csbneggOLD 6521  fsum0diaglem2 8519  fsum0diag 8520

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454  df-csb 2541

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