Theorem r19.27m 3316

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 3310	. . 3 |- ( E. x x e. A -> ( ps <-> A. x e. A ps ) )
3	2	anbi2d 437	. 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 2441	. 2 |- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
5	3, 4	syl6rbbr 188	1 |- ( E. x x e. A -> ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   F/wnf 1349   E.wex 1381   e. wcel 1393   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036  df-ral 2311

This theorem is referenced by:  r19.27mv  3317  raaanlem  3326