Theorem dedhb 2933

Description: A deduction theorem for converting the inference |- F/_ x A => |- ph into a closed theorem. Use nfa1 1555 and nfab 2344 to eliminate the hypothesis of the substitution instance ps of the inference. For converting the inference form into a deduction form, abidnf 2932 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 2932	. . . 4 |- ( F/_ x A -> { z | A. x z e. A } = A )
3	2	eqcomd 2202	. . 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 1362   = wceq 1364   e. wcel 2167   {cab 2182   F/_wnfc 2326

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328

This theorem is referenced by: (None)