Metamath Proof Explorer


Theorem sb4bOLD

Description: Obsolete version of sb4b as of 21-Feb-2024. (Contributed by NM, 27-May-1997) Revise df-sb . (Revised by Wolf Lammen, 25-Jul-2023) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Assertion sb4bOLD
|- ( -. A. x x = t -> ( [ t / x ] ph <-> A. x ( x = t -> ph ) ) )

Proof

Step Hyp Ref Expression
1 nfeqf2
 |-  ( -. A. x x = t -> F/ x y = t )
2 nfnf1
 |-  F/ x F/ x y = t
3 id
 |-  ( F/ x y = t -> F/ x y = t )
4 2 3 nfan1
 |-  F/ x ( F/ x y = t /\ y = t )
5 equequ2
 |-  ( y = t -> ( x = y <-> x = t ) )
6 5 imbi1d
 |-  ( y = t -> ( ( x = y -> ph ) <-> ( x = t -> ph ) ) )
7 6 adantl
 |-  ( ( F/ x y = t /\ y = t ) -> ( ( x = y -> ph ) <-> ( x = t -> ph ) ) )
8 4 7 albid
 |-  ( ( F/ x y = t /\ y = t ) -> ( A. x ( x = y -> ph ) <-> A. x ( x = t -> ph ) ) )
9 8 pm5.74da
 |-  ( F/ x y = t -> ( ( y = t -> A. x ( x = y -> ph ) ) <-> ( y = t -> A. x ( x = t -> ph ) ) ) )
10 1 9 syl
 |-  ( -. A. x x = t -> ( ( y = t -> A. x ( x = y -> ph ) ) <-> ( y = t -> A. x ( x = t -> ph ) ) ) )
11 10 albidv
 |-  ( -. A. x x = t -> ( A. y ( y = t -> A. x ( x = y -> ph ) ) <-> A. y ( y = t -> A. x ( x = t -> ph ) ) ) )
12 df-sb
 |-  ( [ t / x ] ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
13 ax6ev
 |-  E. y y = t
14 13 a1bi
 |-  ( A. x ( x = t -> ph ) <-> ( E. y y = t -> A. x ( x = t -> ph ) ) )
15 19.23v
 |-  ( A. y ( y = t -> A. x ( x = t -> ph ) ) <-> ( E. y y = t -> A. x ( x = t -> ph ) ) )
16 14 15 bitr4i
 |-  ( A. x ( x = t -> ph ) <-> A. y ( y = t -> A. x ( x = t -> ph ) ) )
17 11 12 16 3bitr4g
 |-  ( -. A. x x = t -> ( [ t / x ] ph <-> A. x ( x = t -> ph ) ) )