Theorem bj-wnf2 34900

Description: When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)

Assertion

Ref	Expression
bj-wnf2	⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem bj-wnf2

Step	Hyp	Ref	Expression
1		hbe1 2139	. 2 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2		bj-eximcom 34824	. 2 ⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥∃𝑥𝜑 → ∃𝑥∀𝑥𝜓))
3		hbe1a 2140	. 2 ⊢ (∃𝑥∀𝑥𝜓 → ∀𝑥𝜓)
4	1, 2, 3	syl56 36	1 ⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-10 2137

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  bj-wnfnf  34921