Theorem bj-nnfbd 37435

Description: If two formulas are equivalent, then nonfreeness of a variable in one of them is equivalent to nonfreeness in the other, deduction form. See bj-nnfbi 37413. (Contributed by BJ, 27-Aug-2023.)

Hypothesis

Ref	Expression
bj-nnfbd.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
bj-nnfbd	⊢ (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem bj-nnfbd

Step	Hyp	Ref	Expression
1		ax-5 1943	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		bj-nnfbd.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	bj-nnfbd0 37414	. 2 ⊢ ((𝜑 ∧ ∀𝑥𝜑) → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
4	1, 3	mpdan 700	1 ⊢ (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎ'wnnf 37392

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

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-bj-nnf 37393

This theorem is used by: (None)