Theorem rspn0 3934

Description: Specialization for restricted generalization with a nonempty set. (Contributed by Alexander van der Vekens, 6-Sep-2018.)

Assertion

Ref	Expression
rspn0	|- ( A =/= (/) -> ( A. x e. A ph -> ph ) )

Distinct variable groups:   x, A   ph, x

Proof of Theorem rspn0

Step	Hyp	Ref	Expression
1		n0 3931	. 2 |- ( A =/= (/) <-> E. x x e. A )
2		nfra1 2941	. . . 4 |- F/ x A. x e. A ph
3		nfv 1843	. . . 4 |- F/ x ph
4	2, 3	nfim 1825	. . 3 |- F/ x ( A. x e. A ph -> ph )
5		rsp 2929	. . . 4 |- ( A. x e. A ph -> ( x e. A -> ph ) )
6	5	com12 32	. . 3 |- ( x e. A -> ( A. x e. A ph -> ph ) )
7	4, 6	exlimi 2086	. 2 |- ( E. x x e. A -> ( A. x e. A ph -> ph ) )
8	1, 7	sylbi 207	1 |- ( A =/= (/) -> ( A. x e. A ph -> ph ) )

Syntax hints:   -> wi 4   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  hashge2el2dif  13262  rmodislmodlem  18930  rmodislmod  18931  scmatf1  20337  fusgrregdegfi  26465  rusgr1vtxlem  26483  upgrewlkle2  26502  ralralimp  41295