Theorem r19.28mv 3617

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 1581	. 2 |- F/ x ph
2	1	r19.28m 3614	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   E.wex 1545   e. wcel 2209   A.wral 2528

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-cleq 2231  df-clel 2234  df-ral 2533

This theorem is referenced by:  iinrabm  4070  iindif2m  4075  iinin2m  4076  xpiindim  4912  fintm  5572  ixpiinm  6996  neipsm  15178