Theorem dedhb 3376

Description: A deduction theorem for converting the inference |- F/_ x A => |- ph into a closed theorem. Use nfa1 2028 and nfab 2769 to eliminate the hypothesis of the substitution instance ps of the inference. For converting the inference form into a deduction form, abidnf 3375 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 3375	. . . 4 |- ( F/_ x A -> { z | A. x z e. A } = A )
3	2	eqcomd 2628	. . 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 17	. 2 |- ( F/_ x A -> ( ph <-> ps ) )
6	1, 5	mpbiri 248	1 |- ( F/_ x A -> ph )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751

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

This theorem is referenced by: (None)