Theorem nf3and 1583

Description: Deduction form of bound-variable hypothesis builder nf3an 1580. (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 982	. 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 1582	. . 3 |- ( ph -> F/ x ( ps /\ ch ) )
5		nfand.3	. . 3 |- ( ph -> F/ x th )
6	4, 5	nfand 1582	. 2 |- ( ph -> F/ x ( ( ps /\ ch ) /\ th ) )
7	1, 6	nfxfrd 1489	1 |- ( ph -> F/ x ( ps /\ ch /\ th ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   F/wnf 1474

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-gen 1463

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1475

This theorem is referenced by: (None)