Theorem r19.23v 1733

Description: Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers.

Assertion

Ref	Expression
r19.23v	|- (A.x e. A (ph -> ps) <-> (E.x e. A ph -> ps))

Distinct variable group:   ps,x

Proof of Theorem r19.23v

Step	Hyp	Ref	Expression
1		impexp 347	. . 3 |- (((x e. A /\ ph) -> ps) <-> (x e. A -> (ph -> ps)))
2	1	albii 996	. 2 |- (A.x((x e. A /\ ph) -> ps) <-> A.x(x e. A -> (ph -> ps)))
3		df-rex 1642	. . . 4 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4	3	imbi1i 186	. . 3 |- ((E.x e. A ph -> ps) <-> (E.x(x e. A /\ ph) -> ps))
5		19.23v 1288	. . 3 |- (A.x((x e. A /\ ph) -> ps) <-> (E.x(x e. A /\ ph) -> ps))
6	4, 5	bitr4 176	. 2 |- ((E.x e. A ph -> ps) <-> A.x((x e. A /\ ph) -> ps))
7		df-ral 1641	. 2 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
8	2, 6, 7	3bitr4r 184	1 |- (A.x e. A (ph -> ps) <-> (E.x e. A ph -> ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   e. wcel 955  E.wex 977  A.wral 1637  E.wrex 1638

This theorem is referenced by:  reluni 3255  funimass4 3748  ac6lem 4726  kmlem12 4748  lmuni 7886

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-ral 1641  df-rex 1642

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