Theorem r19.22 1731

Description: Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.)

Assertion

Ref	Expression
r19.22	|- (A.x e. A (ph -> ps) -> (E.x e. A ph -> E.x e. A ps))

Proof of Theorem r19.22

Step	Hyp	Ref	Expression
1		imdistan 442	. . . 4 |- ((x e. A -> (ph -> ps)) <-> ((x e. A /\ ph) -> (x e. A /\ ps)))
2	1	albii 999	. . 3 |- (A.x(x e. A -> (ph -> ps)) <-> A.x((x e. A /\ ph) -> (x e. A /\ ps)))
3		19.22 1039	. . 3 |- (A.x((x e. A /\ ph) -> (x e. A /\ ps)) -> (E.x(x e. A /\ ph) -> E.x(x e. A /\ ps)))
4	2, 3	sylbi 199	. 2 |- (A.x(x e. A -> (ph -> ps)) -> (E.x(x e. A /\ ph) -> E.x(x e. A /\ ps)))
5		df-ral 1649	. 2 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
6		df-rex 1650	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
7		df-rex 1650	. . 3 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
8	6, 7	imbi12i 188	. 2 |- ((E.x e. A ph -> E.x e. A ps) <-> (E.x(x e. A /\ ph) -> E.x(x e. A /\ ps)))
9	4, 5, 8	3imtr4 219	1 |- (A.x e. A (ph -> ps) -> (E.x e. A ph -> E.x e. A 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:  r19.22i 1732  r19.22d 1735  negeu 5355  receu 5701

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

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