Theorem csbexg 2548

Description: The existence of proper substitution into a class.

Assertion

Ref	Expression
csbexg	|- ((A e. C /\ A.x B e. D) -> [_A / x]_B e. _V)

Proof of Theorem csbexg

Step	Hyp	Ref	Expression
1		a4sbc 2457	. . . . 5 |- (A e. C -> (A.x{y | y e. B} e. _V -> [A / x]{y | y e. B} e. _V))
2		elisset 2299	. . . . . . 7 |- (B e. D -> B e. _V)
3		abid2 2011	. . . . . . 7 |- {y | y e. B} = B
4	2, 3	syl5eqel 1975	. . . . . 6 |- (B e. D -> {y | y e. B} e. _V)
5	4	alimi 1338	. . . . 5 |- (A.x B e. D -> A.x{y | y e. B} e. _V)
6	1, 5	syl5 20	. . . 4 |- (A e. C -> (A.x B e. D -> [A / x]{y | y e. B} e. _V))
7	6	imp 377	. . 3 |- ((A e. C /\ A.x B e. D) -> [A / x]{y | y e. B} e. _V)
8		ax-17 1317	. . . . 5 |- (y e. _V -> A.x y e. _V)
9	8	sbcabel 2535	. . . 4 |- (A e. C -> ([A / x]{y | y e. B} e. _V <-> {y | [A / x]y e. B} e. _V))
10	9	adantr 425	. . 3 |- ((A e. C /\ A.x B e. D) -> ([A / x]{y | y e. B} e. _V <-> {y | [A / x]y e. B} e. _V))
11	7, 10	mpbid 212	. 2 |- ((A e. C /\ A.x B e. D) -> {y | [A / x]y e. B} e. _V)
12		df-csb 2541	. 2 |- [_A / x]_B = {y | [A / x]y e. B}
13	11, 12	syl5eqel 1975	1 |- ((A e. C /\ A.x B e. D) -> [_A / x]_B e. _V)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300  [wsbc 1534  {cab 1871  _Vcvv 2292  [_csb 2540

This theorem is referenced by:  csbex 2549  csbexOLD 2550  csbnestglem 2580  csbnestg 2581  csbnest1g 2582  sbcnestg 2583  unirep 15664

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

Copyright terms: Public domain