Theorem rspcsbela 3196

Description: Special case related to rspsbc 3125. (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 3125	. . 3 |- (A e. B -> (A.x e. B C e. D -> [.A / x ].C e. D))
2		sbcel1g 3156	. . 3 |- (A e. B -> ( [.A / x ].C e. D <-> [_A / x]_C e. D))
3	1, 2	sylibd 205	. 2 |- (A e. B -> (A.x e. B C e. D -> [_A / x]_C e. D))
4	3	imp 418	1 |- ((A e. B /\ A.x e. B C e. D) -> [_A / x]_C e. D)

Syntax hints:   -> wi 4   /\ wa 358   e. wcel 1710  A.wral 2615   [.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-ral 2620  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)