Theorem csbexg 3147

Description: The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Assertion

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

Proof of Theorem csbexg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3138	. 2 |- [_A / x]_B = {y | [.A / x ].y e. B}
2		abid2 2471	. . . . . . 7 |- {y | y e. B} = B
3		elex 2868	. . . . . . 7 |- (B e. W -> B e. _V)
4	2, 3	syl5eqel 2437	. . . . . 6 |- (B e. W -> {y | y e. B} e. _V)
5	4	alimi 1559	. . . . 5 |- (A.x B e. W -> A.x{y | y e. B} e. _V)
6		spsbc 3059	. . . . 5 |- (A e. V -> (A.x{y | y e. B} e. _V -> [.A / x ].{y | y e. B} e. _V))
7	5, 6	syl5 28	. . . 4 |- (A e. V -> (A.x B e. W -> [.A / x ].{y | y e. B} e. _V))
8	7	imp 418	. . 3 |- ((A e. V /\ A.x B e. W) -> [.A / x ].{y | y e. B} e. _V)
9		nfcv 2490	. . . . 5 |- F/_x_V
10	9	sbcabel 3124	. . . 4 |- (A e. V -> ( [.A / x ].{y | y e. B} e. _V <-> {y | [.A / x ].y e. B} e. _V))
11	10	adantr 451	. . 3 |- ((A e. V /\ A.x B e. W) -> ( [.A / x ].{y | y e. B} e. _V <-> {y | [.A / x ].y e. B} e. _V))
12	8, 11	mpbid 201	. 2 |- ((A e. V /\ A.x B e. W) -> {y | [.A / x ].y e. B} e. _V)
13	1, 12	syl5eqel 2437	1 |- ((A e. V /\ A.x B e. W) -> [_A / x]_B e. _V)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   e. wcel 1710  {cab 2339  _Vcvv 2860   [.wsbc 3047  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  csbex  3148