Theorem bj-wnf1 37543

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-wnf1	⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem bj-wnf1

Step	Hyp	Ref	Expression
1		bj-modal4e 37541	. . 3 ⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑)
2		hba1 2326	. . 3 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓)
3	1, 2	imim12i 63	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥∃𝑥𝜑 → ∀𝑥∀𝑥𝜓))
4		19.38 1872	. 2 ⊢ ((∃𝑥∃𝑥𝜑 → ∀𝑥∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
5	3, 4	syl 18	1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃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-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-12 2213

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  bj-wnfanf  37545  bj-wnfenf  37546  bj-wnfnf  37607