Theorem r19.22 1272

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 339	. . . 4 |- ((x e. A -> (ph -> ps)) <-> ((x e. A /\ ph) -> (x e. A /\ ps)))
2	1	bial 695	. . 3 |- (A.x(x e. A -> (ph -> ps)) <-> A.x((x e. A /\ ph) -> (x e. A /\ ps)))
3		19.22 722	. . 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 174	. 2 |- (A.x(x e. A -> (ph -> ps)) -> (E.x(x e. A /\ ph) -> E.x(x e. A /\ ps)))
5		df-ral 1205	. 2 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
6		df-rex 1206	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
7		df-rex 1206	. . 3 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
8	6, 7	imbi12i 163	. 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 192	1 |- (A.x e. A (ph -> ps) -> (E.x e. A ph -> E.x e. A 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:  r19.22i 1273  r19.22d 1276  negeu 4124  receu 4215

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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