Metamath Proof Explorer

Theorem sb1OLD

Description: Obsolete version of sb1 as of 21-Feb-2024. (Contributed by NM, 13-May-1993) Revise df-sb . (Revised by Wolf Lammen, 29-Jul-2023) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion sb1OLD 
|- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )

Proof

Step Hyp Ref Expression
1 sbequ2 
 |-  ( x = y -> ( [ y / x ] ph -> ph ) )
2 19.8a 
 |-  ( ( x = y /\ ph ) -> E. x ( x = y /\ ph ) )
3 2 ex 
 |-  ( x = y -> ( ph -> E. x ( x = y /\ ph ) ) )
4 1 3 syld 
 |-  ( x = y -> ( [ y / x ] ph -> E. x ( x = y /\ ph ) ) )
5 4 sps 
 |-  ( A. x x = y -> ( [ y / x ] ph -> E. x ( x = y /\ ph ) ) )
6 sb4b 
 |-  ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )
7 equs4 
 |-  ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
8 6 7 syl6bi 
 |-  ( -. A. x x = y -> ( [ y / x ] ph -> E. x ( x = y /\ ph ) ) )
9 5 8 pm2.61i 
 |-  ( [ y / x ] ph -> E. x ( x = y /\ ph ) )