Theorem r19.29 1295

Description: Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers.

Assertion

Ref	Expression
r19.29	|- ((A.x e. A ph /\ E.x e. A ps) -> E.x e. A (ph /\ ps))

Proof of Theorem r19.29

Step	Hyp	Ref	Expression
1		19.29 752	. . 3 |- ((A.x(x e. A -> ph) /\ E.x(x e. A /\ ps)) -> E.x((x e. A -> ph) /\ (x e. A /\ ps)))
2		anandi 392	. . . . 5 |- ((x e. A /\ (ph /\ ps)) <-> ((x e. A /\ ph) /\ (x e. A /\ ps)))
3		abai 366	. . . . . . 7 |- ((x e. A /\ ph) <-> (x e. A /\ (x e. A -> ph)))
4	3	anbi1i 368	. . . . . 6 |- (((x e. A /\ ph) /\ (x e. A /\ ps)) <-> ((x e. A /\ (x e. A -> ph)) /\ (x e. A /\ ps)))
5		anandi 392	. . . . . 6 |- ((x e. A /\ ((x e. A -> ph) /\ ps)) <-> ((x e. A /\ (x e. A -> ph)) /\ (x e. A /\ ps)))
6	4, 5	bitr4 154	. . . . 5 |- (((x e. A /\ ph) /\ (x e. A /\ ps)) <-> (x e. A /\ ((x e. A -> ph) /\ ps)))
7		an12 370	. . . . 5 |- ((x e. A /\ ((x e. A -> ph) /\ ps)) <-> ((x e. A -> ph) /\ (x e. A /\ ps)))
8	2, 6, 7	3bitr 155	. . . 4 |- ((x e. A /\ (ph /\ ps)) <-> ((x e. A -> ph) /\ (x e. A /\ ps)))
9	8	biex 733	. . 3 |- (E.x(x e. A /\ (ph /\ ps)) <-> E.x((x e. A -> ph) /\ (x e. A /\ ps)))
10	1, 9	sylibr 175	. 2 |- ((A.x(x e. A -> ph) /\ E.x(x e. A /\ ps)) -> E.x(x e. A /\ (ph /\ ps)))
11		df-ral 1205	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
12		df-rex 1206	. . 3 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
13	11, 12	anbi12i 369	. 2 |- ((A.x e. A ph /\ E.x e. A ps) <-> (A.x(x e. A -> ph) /\ E.x(x e. A /\ ps)))
14		df-rex 1206	. 2 |- (E.x e. A (ph /\ ps) <-> E.x(x e. A /\ (ph /\ ps)))
15	10, 13, 14	3imtr4 192	1 |- ((A.x e. A ph /\ E.x e. A ps) -> E.x e. A (ph /\ ps))

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

This theorem is referenced by:  r19.29r 1296

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-gen 677

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

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