Theorem r19.28zv 2354

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 groups:   x,A   ph,x

Proof of Theorem r19.28zv

Step	Hyp	Ref	Expression
1		r19.3rzv 2352	. . 3 |- (A =/= (/) -> (ph <-> A.x e. A ph))
2	1	anbi1d 619	. 2 |- (A =/= (/) -> ((ph /\ A.x e. A ps) <-> (A.x e. A ph /\ A.x e. A ps)))
3		r19.26 1753	. 2 |- (A.x e. A (ph /\ ps) <-> (A.x e. A ph /\ A.x e. A ps))
4	2, 3	syl6rbbr 541	1 |- (A =/= (/) -> (A.x e. A (ph /\ ps) <-> (ph /\ A.x e. A ps)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   =/= wne 1588  A.wral 1648  (/)c0 2283

This theorem is referenced by:  raaan 2364  iindif2 2616  neips 7724  cncnp2 7776

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-v 1815  df-dif 2052  df-nul 2284

Copyright terms: Public domain