Theorem sb4a

Description: A version of one implication of sb4b that does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sb4av when possible. (Contributed by NM, 2-Feb-2007) Revise df-sb . (Revised by Wolf Lammen, 28-Jul-2023) (New usage is discouraged.)

	Ref	Expression
Assertion	sb4a	|- ( [ t / x ] A. t ph -> A. x ( x = t -> ph ) )

Proof

Step	Hyp	Ref	Expression
1		sbequ2	|- ( x = t -> ( [ t / x ] A. t ph -> A. t ph ) )
2	1	sps	|- ( A. x x = t -> ( [ t / x ] A. t ph -> A. t ph ) )
3		axc11r	|- ( A. x x = t -> ( A. t ph -> A. x ph ) )
4		ala1	|- ( A. x ph -> A. x ( x = t -> ph ) )
5	3 4	syl6	|- ( A. x x = t -> ( A. t ph -> A. x ( x = t -> ph ) ) )
6	2 5	syld	|- ( A. x x = t -> ( [ t / x ] A. t ph -> A. x ( x = t -> ph ) ) )
7		sb4b	|- ( -. A. x x = t -> ( [ t / x ] A. t ph <-> A. x ( x = t -> A. t ph ) ) )
8		sp	|- ( A. t ph -> ph )
9	8	imim2i	|- ( ( x = t -> A. t ph ) -> ( x = t -> ph ) )
10	9	alimi	|- ( A. x ( x = t -> A. t ph ) -> A. x ( x = t -> ph ) )
11	7 10	biimtrdi	|- ( -. A. x x = t -> ( [ t / x ] A. t ph -> A. x ( x = t -> ph ) ) )
12	6 11	pm2.61i	|- ( [ t / x ] A. t ph -> A. x ( x = t -> ph ) )