Theorem csbexg 4792

Description: The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 17-Aug-2018.)

Assertion

Ref	Expression
csbexg	|- ( 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 3534	. . 3 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		abid2 2745	. . . . . . . 8 |- { y | y e. B } = B
3		elex 3212	. . . . . . . 8 |- ( B e. W -> B e. _V )
4	2, 3	syl5eqel 2705	. . . . . . 7 |- ( B e. W -> { y | y e. B } e. _V )
5	4	alimi 1739	. . . . . 6 |- ( A. x B e. W -> A. x { y | y e. B } e. _V )
6		spsbc 3448	. . . . . 6 |- ( A e. _V -> ( A. x { y | y e. B } e. _V -> [. A / x ]. { y | y e. B } e. _V ) )
7	5, 6	syl5 34	. . . . 5 |- ( A e. _V -> ( A. x B e. W -> [. A / x ]. { y | y e. B } e. _V ) )
8		nfcv 2764	. . . . . 6 |- F/_ x _V
9	8	sbcabel 3517	. . . . 5 |- ( A e. _V -> ( [. A / x ]. { y | y e. B } e. _V <-> { y | [. A / x ]. y e. B } e. _V ) )
10	7, 9	sylibd 229	. . . 4 |- ( A e. _V -> ( A. x B e. W -> { y | [. A / x ]. y e. B } e. _V ) )
11	10	imp 445	. . 3 |- ( ( A e. _V /\ A. x B e. W ) -> { y | [. A / x ]. y e. B } e. _V )
12	1, 11	syl5eqel 2705	. 2 |- ( ( A e. _V /\ A. x B e. W ) -> [_ A / x ]_ B e. _V )
13		csbprc 3980	. . . 4 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
14		0ex 4790	. . . 4 |- (/) e. _V
15	13, 14	syl6eqel 2709	. . 3 |- ( -. A e. _V -> [_ A / x ]_ B e. _V )
16	15	adantr 481	. 2 |- ( ( -. A e. _V /\ A. x B e. W ) -> [_ A / x ]_ B e. _V )
17	12, 16	pm2.61ian 831	1 |- ( A. x B e. W -> [_ A / x ]_ B e. _V )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   _Vcvv 3200   [.wsbc 3435   [_csb 3533   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by:  csbex  4793  abfmpeld  29454