Theorem bj-rspg 14679

Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2840 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 14678	. 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 1451	1 |- ( A. x e. B ph -> ( A e. B -> ps ) )

Syntax hints:   -> wi 4   = wceq 1353   F/wnf 1460   e. wcel 2148   F/_wnfc 2306   A.wral 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741

This theorem is referenced by:  bj-bdfindisg  14840  bj-findisg  14872