Theorem r19.27m 3397

Description: Restricted quantifier version of Theorem 19.27 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
r19.27m.1	|- F/ x ps

Assertion

Ref	Expression
r19.27m	|- ( E. x x e. A -> ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) )

Distinct variable group:   x, A

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

Proof of Theorem r19.27m

Step	Hyp	Ref	Expression
1		r19.27m.1	. . . 4 |- F/ x ps
2	1	r19.3rm 3390	. . 3 |- ( E. x x e. A -> ( ps <-> A. x e. A ps ) )
3	2	anbi2d 453	. 2 |- ( E. x x e. A -> ( ( A. x e. A ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) ) )
4		r19.26 2511	. 2 |- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
5	3, 4	syl6rbbr 198	1 |- ( E. x x e. A -> ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   F/wnf 1401   E.wex 1433   e. wcel 1445   A.wral 2370

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-cleq 2088  df-clel 2091  df-ral 2375

This theorem is referenced by:  r19.27mv  3398  raaanlem  3407