Theorem dedhb 2972

Description: A deduction theorem for converting the inference |- F/_ x A => |- ph into a closed theorem. Use nfa1 1587 and nfab 2377 to eliminate the hypothesis of the substitution instance ps of the inference. For converting the inference form into a deduction form, abidnf 2971 is useful. (Contributed by NM, 8-Dec-2006.)

Hypotheses

Ref	Expression
dedhb.1	|- ( A = { z | A. x z e. A } -> ( ph <-> ps ) )
dedhb.2	|- ps

Assertion

Ref	Expression
dedhb	|- ( F/_ x A -> ph )

Distinct variable groups:   x, z   z, A

Allowed substitution hints:   ph( x, z)   ps( x, z)   A( x)

Proof of Theorem dedhb

Step	Hyp	Ref	Expression
1		dedhb.2	. 2 |- ps
2		abidnf 2971	. . . 4 |- ( F/_ x A -> { z | A. x z e. A } = A )
3	2	eqcomd 2235	. . 3 |- ( F/_ x A -> A = { z | A. x z e. A } )
4		dedhb.1	. . 3 |- ( A = { z | A. x z e. A } -> ( ph <-> ps ) )
5	3, 4	syl 14	. 2 |- ( F/_ x A -> ( ph <-> ps ) )
6	1, 5	mpbiri 168	1 |- ( F/_ x A -> ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   e. wcel 2200   {cab 2215   F/_wnfc 2359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361

This theorem is referenced by: (None)