Theorem hbalw

Description: Weak version of hbal . Uses only Tarski's FOL axiom schemes. Unlike hbal , this theorem requires that x and y be distinct, i.e., not be bundled. (Contributed by NM, 19-Apr-2017)

	Ref	Expression
Hypotheses	hbalw.1	|- ( x = z -> ( ph <-> ps ) )
	hbalw.2	|- ( ph -> A. x ph )
Assertion	hbalw	|- ( A. y ph -> A. x A. y ph )

Proof

Step	Hyp	Ref	Expression
1		hbalw.1	|- ( x = z -> ( ph <-> ps ) )
2		hbalw.2	|- ( ph -> A. x ph )
3	2	alimi	|- ( A. y ph -> A. y A. x ph )
4	1	alcomiw	|- ( A. y A. x ph -> A. x A. y ph )
5	3 4	syl	|- ( A. y ph -> A. x A. y ph )