Theorem r19.28m 3483

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)

Hypothesis

Ref	Expression
r19.28m.1	|- F/ x ph

Assertion

Ref	Expression
r19.28m	|- ( E. x x e. A -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem r19.28m

Step	Hyp	Ref	Expression
1		r19.26 2583	. 2 |- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
2		r19.28m.1	. . . 4 |- F/ x ph
3	2	r19.3rm 3482	. . 3 |- ( E. x x e. A -> ( ph <-> A. x e. A ph ) )
4	3	anbi1d 461	. 2 |- ( E. x x e. A -> ( ( ph /\ A. x e. A ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) ) )
5	1, 4	bitr4id 198	1 |- ( E. x x e. A -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   F/wnf 1440   E.wex 1472   e. wcel 2128   A.wral 2435

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-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-cleq 2150  df-clel 2153  df-ral 2440

This theorem is referenced by:  r19.28mv  3486  raaanlem  3499