Theorem bj-wnfanf 37079

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

Assertion

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

Proof of Theorem bj-wnfanf

Step	Hyp	Ref	Expression
1		bj-wnf1 37077	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
2		bj-19.23bit 37049	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜓))
3	1, 2	sylg 1831	1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1546  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-10 2154  ax-12 2191

This theorem depends on definitions:  df-bi 209  df-or 855  df-ex 1788  df-nf 1792

This theorem is referenced by: (None)