Metamath Proof Explorer


Theorem nfan

Description: If x is not free in ph and ps , then it is not free in ( ph /\ ps ) . (Contributed by Mario Carneiro, 11-Aug-2016) (Proof shortened by Wolf Lammen, 13-Jan-2018) (Proof shortened by Wolf Lammen, 9-Oct-2021)

Ref Expression
Hypotheses nfan.1
|- F/ x ph
nfan.2
|- F/ x ps
Assertion nfan
|- F/ x ( ph /\ ps )

Proof

Step Hyp Ref Expression
1 nfan.1
 |-  F/ x ph
2 nfan.2
 |-  F/ x ps
3 1 a1i
 |-  ( T. -> F/ x ph )
4 2 a1i
 |-  ( T. -> F/ x ps )
5 3 4 nfand
 |-  ( T. -> F/ x ( ph /\ ps ) )
6 5 mptru
 |-  F/ x ( ph /\ ps )