Theorem r19.29 1756

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 1071	. . 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 510	. . . . 5 |- ((x e. A /\ (ph /\ ps)) <-> ((x e. A /\ ph) /\ (x e. A /\ ps)))
3		abai 479	. . . . . . 7 |- ((x e. A /\ ph) <-> (x e. A /\ (x e. A -> ph)))
4	3	anbi1i 481	. . . . . 6 |- (((x e. A /\ ph) /\ (x e. A /\ ps)) <-> ((x e. A /\ (x e. A -> ph)) /\ (x e. A /\ ps)))
5		anandi 510	. . . . . 6 |- ((x e. A /\ ((x e. A -> ph) /\ ps)) <-> ((x e. A /\ (x e. A -> ph)) /\ (x e. A /\ ps)))
6	4, 5	bitr4 176	. . . . 5 |- (((x e. A /\ ph) /\ (x e. A /\ ps)) <-> (x e. A /\ ((x e. A -> ph) /\ ps)))
7		an12 484	. . . . 5 |- ((x e. A /\ ((x e. A -> ph) /\ ps)) <-> ((x e. A -> ph) /\ (x e. A /\ ps)))
8	2, 6, 7	3bitr 177	. . . 4 |- ((x e. A /\ (ph /\ ps)) <-> ((x e. A -> ph) /\ (x e. A /\ ps)))
9	8	exbii 1051	. . 3 |- (E.x(x e. A /\ (ph /\ ps)) <-> E.x((x e. A -> ph) /\ (x e. A /\ ps)))
10	1, 9	sylibr 200	. 2 |- ((A.x(x e. A -> ph) /\ E.x(x e. A /\ ps)) -> E.x(x e. A /\ (ph /\ ps)))
11		df-ral 1649	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
12		df-rex 1650	. . 3 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
13	11, 12	anbi12i 482	. 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 1650	. 2 |- (E.x e. A (ph /\ ps) <-> E.x(x e. A /\ (ph /\ ps)))
15	10, 13, 14	3imtr4 219	1 |- ((A.x e. A ph /\ E.x e. A ps) -> E.x e. A (ph /\ ps))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958  E.wex 980  A.wral 1645  E.wrex 1646

This theorem is referenced by:  r19.29r 1757  iunfiOLD 4569  ivthlem6 7286  ivthlem7 7287

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-ral 1649  df-rex 1650

Copyright terms: Public domain