Theorem r19.23v 1282

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(s):   ps,x

Proof of Theorem r19.23v

Step	Hyp	Ref	Expression
1		impexp 276	. . 3 |- (((x e. A /\ ph) -> ps) <-> (x e. A -> (ph -> ps)))
2	1	bial 695	. 2 |- (A.x((x e. A /\ ph) -> ps) <-> A.x(x e. A -> (ph -> ps)))
3		df-rex 1206	. . . 4 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4	3	imbi1i 161	. . 3 |- ((E.x e. A ph -> ps) <-> (E.x(x e. A /\ ph) -> ps))
5		19.23v 950	. . 3 |- (A.x((x e. A /\ ph) -> ps) <-> (E.x(x e. A /\ ph) -> ps))
6	4, 5	bitr4 154	. 2 |- ((E.x e. A ph -> ps) <-> A.x((x e. A /\ ph) -> ps))
7		df-ral 1205	. 2 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
8	2, 6, 7	3bitr4r 159	1 |- (A.x e. A (ph -> ps) <-> (E.x e. A ph -> ps))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   e. wcel 1092  A.wral 1201  E.wrex 1202

This theorem is referenced by:  reluni 2493  ac6lem 3575  kmlem11 3590

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-ral 1205  df-rex 1206

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