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Metamath Proof Explorer

Theorem nfbi

Description: If x is not free in ph and ps , then it is not free in ( ph <-> ps ) . (Contributed by NM, 26-May-1993) (Revised by Mario Carneiro, 11-Aug-2016) (Proof shortened by Wolf Lammen, 2-Jan-2018)

Ref Expression
Hypotheses nf.1 
|- F/ x ph
nf.2 
|- F/ x ps
Assertion nfbi 
|- F/ x ( ph <-> ps )

Proof

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