Theorem rspcsbela 2905

Description: Special case related to rspsbc 2840. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)

Assertion

Ref	Expression
rspcsbela	|- ( ( A e. B /\ A. x e. B C e. D ) -> [_ A / x ]_ C e. D )

Distinct variable groups:   x, B   x, D

Allowed substitution hints:   A( x)   C( x)

Proof of Theorem rspcsbela

Step	Hyp	Ref	Expression
1		rspsbc 2840	. . 3 |- ( A e. B -> ( A. x e. B C e. D -> [. A / x ]. C e. D ) )
2		sbcel1g 2869	. . 3 |- ( A e. B -> ( [. A / x ]. C e. D <-> [_ A / x ]_ C e. D ) )
3	1, 2	sylibd 138	. 2 |- ( A e. B -> ( A. x e. B C e. D -> [_ A / x ]_ C e. D ) )
4	3	imp 115	1 |- ( ( A e. B /\ A. x e. B C e. D ) -> [_ A / x ]_ C e. D )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   A.wral 2306   [.wsbc 2764   [_csb 2852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-sbc 2765  df-csb 2853

This theorem is referenced by: (None)