Theorem r19.28m 3584

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 2659	. 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 3583	. . 3 |- ( E. x x e. A -> ( ph <-> A. x e. A ph ) )
4	3	anbi1d 465	. 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 199	1 |- ( E. x x e. A -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   F/wnf 1508   E.wex 1540   e. wcel 2202   A.wral 2510

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-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-cleq 2224  df-clel 2227  df-ral 2515

This theorem is referenced by:  r19.28mv  3587  raaanlem  3599