Theorem r19.28z 4063

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)

Hypothesis

Ref	Expression
r19.3rz.1	|- F/ x ph

Assertion

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

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem r19.28z

Step	Hyp	Ref	Expression
1		r19.3rz.1	. . . 4 |- F/ x ph
2	1	r19.3rz 4062	. . 3 |- ( A =/= (/) -> ( ph <-> A. x e. A ph ) )
3	2	anbi1d 741	. 2 |- ( A =/= (/) -> ( ( ph /\ A. x e. A ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) ) )
4		r19.26 3064	. 2 |- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
5	3, 4	syl6rbbr 279	1 |- ( A =/= (/) -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   F/wnf 1708   =/= wne 2794   A.wral 2912   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  r19.28zv  4066  raaan  4082  raaan2  41175