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Metamath Proof Explorer

Theorem hb3an

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)

Ref Expression
Hypotheses hb.1 
|- ( ph -> A. x ph )
hb.2 
|- ( ps -> A. x ps )
hb.3 
|- ( ch -> A. x ch )
Assertion hb3an 
|- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) )

Proof

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