Metamath Proof Explorer

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 ⊢ ∃ x ψ → ψ → ∃ x φ → ψ → ∀ x φ → ψ

Proof

Step Hyp Ref Expression
1 bj-eximcom ⊢ ∃ x φ → ψ → ∀ x φ → ∃ x ψ
2 imim2 ⊢ ∃ x ψ → ψ → ∀ x φ → ∃ x ψ → ∀ x φ → ψ
3 1 2 syl5 ⊢ ∃ x ψ → ψ → ∃ x φ → ψ → ∀ x φ → ψ