Theorem bj-rspg 13165

Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2790 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
bj-rspg.nfa	|- F/_ x A
bj-rspg.nfb	|- F/_ x B
bj-rspg.nf2	|- F/ x ps
bj-rspg.is	|- ( x = A -> ( ph -> ps ) )

Assertion

Ref	Expression
bj-rspg	|- ( A. x e. B ph -> ( A e. B -> ps ) )

Proof of Theorem bj-rspg

Step	Hyp	Ref	Expression
1		bj-rspg.nfa	. . 3 |- F/_ x A
2		bj-rspg.nfb	. . 3 |- F/_ x B
3		bj-rspg.nf2	. . 3 |- F/ x ps
4	1, 2, 3	bj-rspgt 13164	. 2 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A. x e. B ph -> ( A e. B -> ps ) ) )
5		bj-rspg.is	. 2 |- ( x = A -> ( ph -> ps ) )
6	4, 5	mpg 1428	1 |- ( A. x e. B ph -> ( A e. B -> ps ) )

Syntax hints:   -> wi 4   = wceq 1332   F/wnf 1437   e. wcel 1481   F/_wnfc 2269   A.wral 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691

This theorem is referenced by:  bj-bdfindisg  13317  bj-findisg  13349