Theorem bj-cbveximd

Description: A lemma for alpha-renaming of variables bound by an existential quantifier. (Contributed by BJ, 4-Apr-2026) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbveximd.nf0	|- ( ph -> A. x ph )
	bj-cbveximd.nf1	|- ( ph -> A. y ph )
	bj-cbveximd.nfch	|- ( ph -> ( ch -> A. y ch ) )
	bj-cbveximd.nfth	|- ( ph -> ( E. x th -> th ) )
	bj-cbveximd.denote	|- ( ph -> A. x E. y ps )
	bj-cbveximd.maj	|- ( ( ph /\ ps ) -> ( ch -> th ) )
Assertion	bj-cbveximd	|- ( ph -> ( E. x ch -> E. y th ) )

Proof

Step	Hyp	Ref	Expression
1		bj-cbveximd.nf0	|- ( ph -> A. x ph )
2		bj-cbveximd.nf1	|- ( ph -> A. y ph )
3		bj-cbveximd.nfch	|- ( ph -> ( ch -> A. y ch ) )
4		bj-cbveximd.nfth	|- ( ph -> ( E. x th -> th ) )
5		bj-cbveximd.denote	|- ( ph -> A. x E. y ps )
6		bj-cbveximd.maj	|- ( ( ph /\ ps ) -> ( ch -> th ) )
7		excomim	|- ( E. x E. y th -> E. y E. x th )
8	2 4	eximdh	|- ( ph -> ( E. y E. x th -> E. y th ) )
9	7 8	syl5	|- ( ph -> ( E. x E. y th -> E. y th ) )
10	1 2 3 9 5 6	bj-cbveximdlem	|- ( ph -> ( E. x ch -> E. y th ) )