Theorem r19.35 2231

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 2219	. . . 4 |- (A.x e. A (ph /\ -. ps) <-> (A.x e. A ph /\ A.x e. A -. ps))
2		annim 257	. . . . 5 |- ((ph /\ -. ps) <-> -. (ph -> ps))
3	2	ralbii 2127	. . . 4 |- (A.x e. A (ph /\ -. ps) <-> A.x e. A -. (ph -> ps))
4		df-an 242	. . . 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	3bitr3i 198	. . 3 |- (A.x e. A -. (ph -> ps) <-> -. (A.x e. A ph -> -. A.x e. A -. ps))
6	5	con2bii 238	. 2 |- ((A.x e. A ph -> -. A.x e. A -. ps) <-> -. A.x e. A -. (ph -> ps))
7		dfrex2 2116	. . 3 |- (E.x e. A ps <-> -. A.x e. A -. ps)
8	7	imbi2i 202	. 2 |- ((A.x e. A ph -> E.x e. A ps) <-> (A.x e. A ph -> -. A.x e. A -. ps))
9		dfrex2 2116	. 2 |- (E.x e. A (ph -> ps) <-> -. A.x e. A -. (ph -> ps))
10	6, 8, 9	3bitr4ri 201	1 |- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wral 2105  E.wrex 2106

This theorem is referenced by:  r19.36av 2232  r19.37av 2233  r19.43 2238  r19.37zv 2965  r19.36zv 2969  bndndx 7282  metcnp4 9248  nmobndseqi 9780  cexint2 14862

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-ral 2109  df-rex 2110

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