Theorem r19.35 1756

Description: Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90.

Assertion

Ref	Expression
r19.35	|- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))

Proof of Theorem r19.35

Step	Hyp	Ref	Expression
1		r19.26 1747	. . . 4 |- (A.x e. A (ph /\ -. ps) <-> (A.x e. A ph /\ A.x e. A -. ps))
2		annim 238	. . . . 5 |- ((ph /\ -. ps) <-> -. (ph -> ps))
3	2	ralbii 1664	. . . 4 |- (A.x e. A (ph /\ -. ps) <-> A.x e. A -. (ph -> ps))
4		df-an 225	. . . 4 |- ((A.x e. A ph /\ A.x e. A -. ps) <-> -. (A.x e. A ph -> -. A.x e. A -. ps))
5	1, 3, 4	3bitr3 181	. . 3 |- (A.x e. A -. (ph -> ps) <-> -. (A.x e. A ph -> -. A.x e. A -. ps))
6	5	con2bii 221	. 2 |- ((A.x e. A ph -> -. A.x e. A -. ps) <-> -. A.x e. A -. (ph -> ps))
7		dfrex2 1653	. . 3 |- (E.x e. A ps <-> -. A.x e. A -. ps)
8	7	imbi2i 185	. 2 |- ((A.x e. A ph -> E.x e. A ps) <-> (A.x e. A ph -> -. A.x e. A -. ps))
9		dfrex2 1653	. 2 |- (E.x e. A (ph -> ps) <-> -. A.x e. A -. (ph -> ps))
10	6, 8, 9	3bitr4r 184	1 |- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wral 1642  E.wrex 1643

This theorem is referenced by:  r19.36av 1757  r19.37av 1758  r19.37zv 2347  r19.36zv 2350  bndndx 6028  metcnp4 7920  nmobndseqi 8385  faimpex 10375

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-4 971  ax-5o 973

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-ral 1646  df-rex 1647

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