Theorem r19.32r 2501

Description: One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence. (Contributed by Jim Kingdon, 19-Aug-2018.)

Hypothesis

Ref	Expression
r19.32r.1	|- F/ x ph

Assertion

Ref	Expression
r19.32r	|- ( ( ph \/ A. x e. A ps ) -> A. x e. A ( ph \/ ps ) )

Proof of Theorem r19.32r

Step	Hyp	Ref	Expression
1		r19.32r.1	. . . 4 |- F/ x ph
2		orc 665	. . . . 5 |- ( ph -> ( ph \/ ps ) )
3	2	a1d 22	. . . 4 |- ( ph -> ( x e. A -> ( ph \/ ps ) ) )
4	1, 3	alrimi 1455	. . 3 |- ( ph -> A. x ( x e. A -> ( ph \/ ps ) ) )
5		df-ral 2353	. . . 4 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
6		olc 664	. . . . . 6 |- ( ps -> ( ph \/ ps ) )
7	6	imim2i 12	. . . . 5 |- ( ( x e. A -> ps ) -> ( x e. A -> ( ph \/ ps ) ) )
8	7	alimi 1384	. . . 4 |- ( A. x ( x e. A -> ps ) -> A. x ( x e. A -> ( ph \/ ps ) ) )
9	5, 8	sylbi 119	. . 3 |- ( A. x e. A ps -> A. x ( x e. A -> ( ph \/ ps ) ) )
10	4, 9	jaoi 668	. 2 |- ( ( ph \/ A. x e. A ps ) -> A. x ( x e. A -> ( ph \/ ps ) ) )
11		df-ral 2353	. 2 |- ( A. x e. A ( ph \/ ps ) <-> A. x ( x e. A -> ( ph \/ ps ) ) )
12	10, 11	sylibr 132	1 |- ( ( ph \/ A. x e. A ps ) -> A. x e. A ( ph \/ ps ) )

Syntax hints:   -> wi 4   \/ wo 661   A.wal 1282   F/wnf 1389   e. wcel 1433   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-gen 1378  ax-4 1440

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-ral 2353

This theorem is referenced by:  r19.32vr  2502