Theorem dedhb 2899

Description: A deduction theorem for converting the inference |- F/_ x A => |- ph into a closed theorem. Use nfa1 1534 and nfab 2317 to eliminate the hypothesis of the substitution instance ps of the inference. For converting the inference form into a deduction form, abidnf 2898 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 2898	. . . 4 |- ( F/_ x A -> { z | A. x z e. A } = A )
3	2	eqcomd 2176	. . 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 1346   = wceq 1348   e. wcel 2141   {cab 2156   F/_wnfc 2299

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301

This theorem is referenced by: (None)