Theorem nfded 34254

Description: A deduction theorem that converts a not-free inference directly to deduction form. The first hypothesis is the hypothesis of the deduction form. The second is an equality deduction (e.g. ( F/_ x A -> U. { y | A. x y e. A } = U. A )) that starts from abidnf 3375. The last is assigned to the inference form (e.g. F/_ x U. { y | A. x y e. A }) whose hypothesis is satisfied using nfaba1 2770. (Contributed by NM, 19-Nov-2020.)

Hypotheses

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

Assertion

Ref	Expression
nfded	|- ( ph -> F/_ x C )

Proof of Theorem nfded

Step	Hyp	Ref	Expression
1		nfded.3	. 2 |- F/_ x B
2		nfded.1	. . 3 |- ( ph -> F/_ x A )
3		nfnfc1 2767	. . . 4 |- F/ x F/_ x A
4		nfded.2	. . . 4 |- ( F/_ x A -> B = C )
5	3, 4	nfceqdf 2760	. . 3 |- ( F/_ x A -> ( F/_ x B <-> F/_ x C ) )
6	2, 5	syl 17	. 2 |- ( ph -> ( F/_ x B <-> F/_ x C ) )
7	1, 6	mpbii 223	1 |- ( ph -> F/_ x C )

Syntax hints:   -> wi 4   <-> wb 196   = 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-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  nfunidALT  34257