Theorem nfded2 34255

Description: A deduction theorem that converts a not-free inference directly to deduction form. The first 2 hypotheses are the hypotheses of the deduction form. The third is an equality deduction (e.g. ( ( F/_ x A /\ F/_ x B ) -> <. { y | A. x y e. A } , { y | A. x y e. B } >. = <. A , B >. ) for nfopd 4419) that starts from abidnf 3375. The last is assigned to the inference form (e.g. F/_ x <. { y | A. x y e. A } , { y | A. x y e. B } >. for nfop 4418) whose hypotheses are satisfied using nfaba1 2770. (Contributed by NM, 19-Nov-2020.)

Hypotheses

Ref	Expression
nfded2.1	|- ( ph -> F/_ x A )
nfded2.2	|- ( ph -> F/_ x B )
nfded2.3	|- ( ( F/_ x A /\ F/_ x B ) -> C = D )
nfded2.4	|- F/_ x C

Assertion

Ref	Expression
nfded2	|- ( ph -> F/_ x D )

Proof of Theorem nfded2

Step	Hyp	Ref	Expression
1		nfded2.4	. 2 |- F/_ x C
2		nfded2.1	. . 3 |- ( ph -> F/_ x A )
3		nfded2.2	. . 3 |- ( ph -> F/_ x B )
4		nfnfc1 2767	. . . . 5 |- F/ x F/_ x A
5		nfnfc1 2767	. . . . 5 |- F/ x F/_ x B
6	4, 5	nfan 1828	. . . 4 |- F/ x ( F/_ x A /\ F/_ x B )
7		nfded2.3	. . . 4 |- ( ( F/_ x A /\ F/_ x B ) -> C = D )
8	6, 7	nfceqdf 2760	. . 3 |- ( ( F/_ x A /\ F/_ x B ) -> ( F/_ x C <-> F/_ x D ) )
9	2, 3, 8	syl2anc 693	. 2 |- ( ph -> ( F/_ x C <-> F/_ x D ) )
10	1, 9	mpbii 223	1 |- ( ph -> F/_ x D )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   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-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-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  nfopdALT  34258