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Metamath Proof Explorer

Theorem sbievOLD

Description: Obsolete version of sbiev as of 24-Aug-2025. (Contributed by NM, 30-Jun-1994) (Revised by Wolf Lammen, 18-Jan-2023) Remove dependence on ax-10 and shorten proof. (Revised by BJ, 18-Jul-2023) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses sbiev.1 
|- F/ x ps
sbiev.2 
|- ( x = y -> ( ph <-> ps ) )
Assertion sbievOLD 
|- ( [ y / x ] ph <-> ps )

Proof

Step Hyp Ref Expression
1 sbiev.1 
 |-  F/ x ps
2 sbiev.2 
 |-  ( x = y -> ( ph <-> ps ) )
3 sb6 
 |-  ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
4 1 2 equsalv 
 |-  ( A. x ( x = y -> ph ) <-> ps )
5 3 4 bitri 
 |-  ( [ y / x ] ph <-> ps )