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Metamath Proof Explorer

Theorem sbnfOLD

Description: Obsolete version of sbnf as of 2-May-2025. (Contributed by BJ, 2-May-2019) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion sbnfOLD 
|- ( [ z / y ] F/ x ph <-> F/ x [ z / y ] ph )

Proof

Step Hyp Ref Expression
1 sbim 
 |-  ( [ z / y ] ( ph -> A. x ph ) <-> ( [ z / y ] ph -> [ z / y ] A. x ph ) )
2 sbal 
 |-  ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )
3 2 imbi2i 
 |-  ( ( [ z / y ] ph -> [ z / y ] A. x ph ) <-> ( [ z / y ] ph -> A. x [ z / y ] ph ) )
4 1 3 bitri 
 |-  ( [ z / y ] ( ph -> A. x ph ) <-> ( [ z / y ] ph -> A. x [ z / y ] ph ) )
5 4 albii 
 |-  ( A. x [ z / y ] ( ph -> A. x ph ) <-> A. x ( [ z / y ] ph -> A. x [ z / y ] ph ) )
6 nf5 
 |-  ( F/ x ph <-> A. x ( ph -> A. x ph ) )
7 6 sbbii 
 |-  ( [ z / y ] F/ x ph <-> [ z / y ] A. x ( ph -> A. x ph ) )
8 sbal 
 |-  ( [ z / y ] A. x ( ph -> A. x ph ) <-> A. x [ z / y ] ( ph -> A. x ph ) )
9 7 8 bitri 
 |-  ( [ z / y ] F/ x ph <-> A. x [ z / y ] ( ph -> A. x ph ) )
10 nf5 
 |-  ( F/ x [ z / y ] ph <-> A. x ( [ z / y ] ph -> A. x [ z / y ] ph ) )
11 5 9 10 3bitr4i 
 |-  ( [ z / y ] F/ x ph <-> F/ x [ z / y ] ph )