Theorem r19.36zv 4072

Description: Restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 20-Sep-2003.)

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

Allowed substitution hint:   ph( x)

Proof of Theorem r19.36zv

Step	Hyp	Ref	Expression
1		r19.9rzv 4065	. . 3 |- ( A =/= (/) -> ( ps <-> E. x e. A ps ) )
2	1	imbi2d 330	. 2 |- ( A =/= (/) -> ( ( A. x e. A ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) ) )
3		r19.35 3084	. 2 |- ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) )
4	2, 3	syl6rbbr 279	1 |- ( A =/= (/) -> ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)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-rex 2918  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)