Theorem bj-wnfenf 37066

Description: When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the existential form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)

Assertion

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

Proof of Theorem bj-wnfenf

Step	Hyp	Ref	Expression
1		bj-wnf1 37063	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
2		bj-19.21bit 37034	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → 𝜓))
3	1, 2	sylg 1830	1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by: (None)