Theorem r19.36zv 2358

Description: Restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty.

Assertion

Ref	Expression
r19.36zv	|- (A =/= (/) -> (E.x e. A (ph -> ps) <-> (A.x e. A ph -> ps)))

Distinct variable groups:   x,A   ps,x

Proof of Theorem r19.36zv

Step	Hyp	Ref	Expression
1		r19.9rzv 2353	. . 3 |- (A =/= (/) -> (ps <-> E.x e. A ps))
2	1	imbi2d 614	. 2 |- (A =/= (/) -> ((A.x e. A ph -> ps) <-> (A.x e. A ph -> E.x e. A ps)))
3		r19.35 1762	. 2 |- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))
4	2, 3	syl6rbbr 541	1 |- (A =/= (/) -> (E.x e. A (ph -> ps) <-> (A.x e. A ph -> ps)))

Syntax hints:   -> wi 3   <-> wb 146   =/= wne 1588  A.wral 1648  E.wrex 1649  (/)c0 2283

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-nul 2284

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