Theorem hb3an 2129

Description: If x is not free in ph, ps, and ch, it is not free in ( ph /\ ps /\ ch ). (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Hypotheses

Ref	Expression
hb.1	|- ( ph -> A. x ph )
hb.2	|- ( ps -> A. x ps )
hb.3	|- ( ch -> A. x ch )

Assertion

Ref	Expression
hb3an	|- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) )

Proof of Theorem hb3an

Step	Hyp	Ref	Expression
1		hb.1	. . . 4 |- ( ph -> A. x ph )
2	1	nf5i 2024	. . 3 |- F/ x ph
3		hb.2	. . . 4 |- ( ps -> A. x ps )
4	3	nf5i 2024	. . 3 |- F/ x ps
5		hb.3	. . . 4 |- ( ch -> A. x ch )
6	5	nf5i 2024	. . 3 |- F/ x ch
7	2, 4, 6	nf3an 1831	. 2 |- F/ x ( ph /\ ps /\ ch )
8	7	nf5ri 2065	1 |- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) )

Syntax hints:   -> wi 4   /\ w3a 1037   A.wal 1481

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-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  bnj982  30849  bnj1276  30885  bnj1350  30896