Theorem r19.28mv 3460

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, 6-Aug-2018.)

Assertion

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

Distinct variable groups:   x, A   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem r19.28mv

Step	Hyp	Ref	Expression
1		nfv 1509	. 2 |- F/ x ph
2	1	r19.28m 3457	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   E.wex 1469   e. wcel 1481   A.wral 2417

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-clel 2136  df-ral 2422

This theorem is referenced by:  iinrabm  3883  iindif2m  3888  iinin2m  3889  xpiindim  4684  fintm  5316  ixpiinm  6626  neipsm  12362