Theorem bj-spimenfa

Description: An existential generalization result: if ph holds and implies ps for at least one value of x , and if furthermore x is A. -weakly nonfree in ph , then ps holds for at least one value of x . (Contributed by BJ, 3-Apr-2026) Proof should not use 19.35 . (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-spimenfa	|- ( ( ph -> A. x ph ) -> ( E. x ( ph -> ps ) -> ( ph -> E. x ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-eximcom	|- ( E. x ( ph -> ps ) -> ( A. x ph -> E. x ps ) )
2		imim1	|- ( ( ph -> A. x ph ) -> ( ( A. x ph -> E. x ps ) -> ( ph -> E. x ps ) ) )
3	1 2	syl5	|- ( ( ph -> A. x ph ) -> ( E. x ( ph -> ps ) -> ( ph -> E. x ps ) ) )