Theorem r19.37zv 2351

Description: Restricted quantifier version of Theorem 19.37 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Paul Chapman, 8-Oct-2007.)

Assertion

Ref	Expression
r19.37zv	|- (A =/= (/) -> (E.x e. A (ph -> ps) <-> (ph -> E.x e. A ps)))

Distinct variable groups:   x,A   ph,x

Proof of Theorem r19.37zv

Step	Hyp	Ref	Expression
1		r19.3rzv 2348	. . 3 |- (A =/= (/) -> (ph <-> A.x e. A ph))
2	1	imbi1d 613	. 2 |- (A =/= (/) -> ((ph -> E.x e. A ps) <-> (A.x e. A ph -> E.x e. A ps)))
3		r19.35 1759	. 2 |- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))
4	2, 3	syl6rbbr 539	1 |- (A =/= (/) -> (E.x e. A (ph -> ps) <-> (ph -> E.x e. A ps)))

Syntax hints:   -> wi 3   <-> wb 146   =/= wne 1585  A.wral 1645  E.wrex 1646  (/)c0 2280

This theorem is referenced by:  ivthlem6 7286  ivthlem7 7287

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-nul 2281

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