Theorem bj-wnfnf 37777

Description: When 𝜑 is substituted for 𝜓, this statement expresses nonfreeness in the weak form of nonfreeness (∃ → ∀). This could also be proved from bj-nnfim 37746, bj-nnfe1 37779 and bj-nnfa1 37778. (Contributed by BJ, 9-Dec-2023.)

Assertion

Ref	Expression
bj-wnfnf	⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)

Proof of Theorem bj-wnfnf

Step	Hyp	Ref	Expression
1		bj-wnf2 37714	. 2 ⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
2		bj-wnf1 37713	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
3		df-bj-nnf 37721	. 2 ⊢ (Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓) ↔ ((∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)) ∧ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))))
4	1, 2, 3	mpbir2an 724	1 ⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811  Ⅎ'wnnf 37720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-bj-nnf 37721

This theorem is used by: (None)