Theorem bj-wnf2 37592

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 2180	. 2 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2		bj-eximcom 37486	. 2 ⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥∃𝑥𝜑 → ∃𝑥∀𝑥𝜓))
3		hbe1a 2181	. 2 ⊢ (∃𝑥∀𝑥𝜓 → ∀𝑥𝜓)
4	1, 2, 3	syl56 37	1 ⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-10 2178

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  bj-wnfnf  37655