Theorem bj-wnfnf 34087

Description: When 𝜑 is substituted for 𝜓, this statement expresses nonfreeness in the weak form of nonfreeness (∃ → ∀). Note that this could also be proved from bj-nnfim 34094, bj-nnfe1 34108 and bj-nnfa1 34107. (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 34071	. 2 ⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
2		bj-wnf1 34070	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
3		df-bj-nnf 34075	. 2 ⊢ (Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓) ↔ ((∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)) ∧ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))))
4	1, 2, 3	mpbir2an 709	1 ⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1534  ∃wex 1779  Ⅎ'wnnf 34074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784  df-bj-nnf 34075

This theorem is referenced by: (None)