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		df-sb	|- ( [ 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 ) ) )