Theorem bj-spimnfe

Description: A universal specification result: if ph is true for all values of x and implies ps for at least one value, and if furthermore x is E. -weakly nonfree in ps , then ps follows. An intermediate result on the way to prove 19.36i , bj-19.36im , 19.36imv , spimfw ... (Contributed by BJ, 3-Apr-2026) Proof should not use 19.35 . (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-eximcom	|- ( E. x ( ph -> ps ) -> ( A. x ph -> E. x ps ) )
2		imim2	|- ( ( E. x ps -> ps ) -> ( ( A. x ph -> E. x ps ) -> ( A. x ph -> ps ) ) )
3	1 2	syl5	|- ( ( E. x ps -> ps ) -> ( E. x ( ph -> ps ) -> ( A. x ph -> ps ) ) )