Theorem bj-nnfnfTEMP 34086

Description: New nonfreeness implies old nonfreeness on implicational calculus (the proof indicates it uses ax-3 8 because of set.mm's definition of the biconditional, but the proof actually holds in minimal implicational calculus). (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1784 except via df-nf 1784 directly. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nnfnfTEMP	⊢ (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑)

Proof of Theorem bj-nnfnfTEMP

Step	Hyp	Ref	Expression
1		bj-nnfea 34083	. 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
2		df-nf 1784	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	sylibr 236	1 ⊢ (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1534  ∃wex 1779  Ⅎwnf 1783  Ⅎ'wnnf 34074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is referenced by: (None)