Theorem dedhb 2895

Description: A deduction theorem for converting the inference |- F/_ x A => |- ph into a closed theorem. Use nfa1 1529 and nfab 2313 to eliminate the hypothesis of the substitution instance ps of the inference. For converting the inference form into a deduction form, abidnf 2894 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 2894	. . . 4 |- ( F/_ x A -> { z | A. x z e. A } = A )
3	2	eqcomd 2171	. . 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 167	1 |- ( F/_ x A -> ph )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   e. wcel 2136   {cab 2151   F/_wnfc 2295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297

This theorem is referenced by: (None)