Theorem r19.27m 3542

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		r19.26 2620	. 2 |- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
2		r19.27m.1	. . . 4 |- F/ x ps
3	2	r19.3rm 3535	. . 3 |- ( E. x x e. A -> ( ps <-> A. x e. A ps ) )
4	3	anbi2d 464	. 2 |- ( E. x x e. A -> ( ( A. x e. A ph /\ 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 ) <-> ( A. x e. A ph /\ ps ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   F/wnf 1471   E.wex 1503   e. wcel 2164   A.wral 2472

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-cleq 2186  df-clel 2189  df-ral 2477

This theorem is referenced by:  r19.27mv  3543  raaanlem  3551