Theorem bj-spimenfa 36887

Description: An existential generalization result: if 𝜑 holds and implies 𝜓 for at least one value of 𝑥, and if furthermore 𝑥 is ∀ -weakly nonfree in 𝜑, then 𝜓 holds for at least one value of 𝑥. (Contributed by BJ, 3-Apr-2026.) Proof should not use 19.35 1879. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-spimenfa	⊢ ((𝜑 → ∀𝑥𝜑) → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))

Proof of Theorem bj-spimenfa

Step	Hyp	Ref	Expression
1		bj-eximcom 36879	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
2		imim1 83	. 2 ⊢ ((𝜑 → ∀𝑥𝜑) → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑 → ∃𝑥𝜓)))
3	1, 2	syl5 34	1 ⊢ ((𝜑 → ∀𝑥𝜑) → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 207  df-ex 1782

This theorem is referenced by:  bj-spime  36889  bj-19.37im  37029