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Metamath Proof Explorer

Theorem sb4b

Description: Simplified definition of substitution when variables are distinct. Version of sb6 with a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 27-May-1997) Revise df-sb . (Revised by Wolf Lammen, 21-Feb-2024) (New usage is discouraged.)

Ref Expression
Assertion sb4b 
|- ( -. A. x x = t -> ( [ t / x ] ph <-> A. x ( x = t -> ph ) ) )

Proof

Step Hyp Ref Expression
1 nfna1 
 |-  F/ x -. A. x x = t
2 nfeqf2 
 |-  ( -. A. x x = t -> F/ x y = t )
3 1 2 nfan1 
 |-  F/ x ( -. A. x x = t /\ y = t )
4 equequ2 
 |-  ( y = t -> ( x = y <-> x = t ) )
5 4 imbi1d 
 |-  ( y = t -> ( ( x = y -> ph ) <-> ( x = t -> ph ) ) )
6 5 adantl 
 |-  ( ( -. A. x x = t /\ y = t ) -> ( ( x = y -> ph ) <-> ( x = t -> ph ) ) )
7 3 6 albid 
 |-  ( ( -. A. x x = t /\ y = t ) -> ( A. x ( x = y -> ph ) <-> A. x ( x = t -> ph ) ) )
8 7 pm5.74da 
 |-  ( -. A. x x = t -> ( ( y = t -> A. x ( x = y -> ph ) ) <-> ( y = t -> A. x ( x = t -> ph ) ) ) )
9 8 albidv 
 |-  ( -. A. x x = t -> ( A. y ( y = t -> A. x ( x = y -> ph ) ) <-> A. y ( y = t -> A. x ( x = t -> ph ) ) ) )
10 dfsb 
 |-  ( [ t / x ] ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
11 ax6ev 
 |-  E. y y = t
12 11 a1bi 
 |-  ( A. x ( x = t -> ph ) <-> ( E. y y = t -> A. x ( x = t -> ph ) ) )
13 19.23v 
 |-  ( A. y ( y = t -> A. x ( x = t -> ph ) ) <-> ( E. y y = t -> A. x ( x = t -> ph ) ) )
14 12 13 bitr4i 
 |-  ( A. x ( x = t -> ph ) <-> A. y ( y = t -> A. x ( x = t -> ph ) ) )
15 9 10 14 3bitr4g 
 |-  ( -. A. x x = t -> ( [ t / x ] ph <-> A. x ( x = t -> ph ) ) )