Theorem nf3and 2019

Description: Deduction form of bound-variable hypothesis builder nf3an 2023. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)

Hypotheses

Ref	Expression
nfand.1	|- ( ph -> F/ x ps )
nfand.2	|- ( ph -> F/ x ch )
nfand.3	|- ( ph -> F/ x th )

Assertion

Ref	Expression
nf3and	|- ( ph -> F/ x ( ps /\ ch /\ th ) )

Proof of Theorem nf3and

Step	Hyp	Ref	Expression
1		df-3an 993	. 2 |- ( ( ps /\ ch /\ th ) <-> ( ( ps /\ ch ) /\ th ) )
2		nfand.1	. . . 4 |- ( ph -> F/ x ps )
3		nfand.2	. . . 4 |- ( ph -> F/ x ch )
4	2, 3	nfand 2018	. . 3 |- ( ph -> F/ x ( ps /\ ch ) )
5		nfand.3	. . 3 |- ( ph -> F/ x th )
6	4, 5	nfand 2018	. 2 |- ( ph -> F/ x ( ( ps /\ ch ) /\ th ) )
7	1, 6	nfxfrd 1707	1 |- ( ph -> F/ x ( ps /\ ch /\ th ) )

Syntax hints:   -> wi 4   /\ wa 375   /\ w3a 991   F/wnf 1677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943

This theorem depends on definitions:  df-bi 190  df-an 377  df-3an 993  df-ex 1674  df-nf 1678

This theorem is referenced by: (None)