Theorem r19.28zv 1769

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty.

Assertion

Ref	Expression
r19.28zv	|- (-. A = (/) -> (A.x e. A (ph /\ ps) <-> (ph /\ A.x e. A ps)))

Distinct variable group(s):   x,A   ph,x

Proof of Theorem r19.28zv

Step	Hyp	Ref	Expression
1		r19.3rzv 1767	. . 3 |- (-. A = (/) -> (ph <-> A.x e. A ph))
2	1	anbi1d 469	. 2 |- (-. A = (/) -> ((ph /\ A.x e. A ps) <-> (A.x e. A ph /\ A.x e. A ps)))
3		r19.26 1289	. 2 |- (A.x e. A (ph /\ ps) <-> (A.x e. A ph /\ A.x e. A ps))
4	2, 3	syl6rbbr 417	1 |- (-. A = (/) -> (A.x e. A (ph /\ ps) <-> (ph /\ A.x e. A ps)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091  A.wral 1201  (/)c0 1707

This theorem is referenced by:  iindif2 2033

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-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-nul 1708

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