Theorem bj-nnfbd 34908

Description: If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other, deduction form. See bj-nnfbi 34907. (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		bj-nnfbd.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	alrimiv 1930	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
3		bj-nnfbi 34907	. 2 ⊢ (((𝜓 ↔ 𝜒) ∧ ∀𝑥(𝜓 ↔ 𝜒)) → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  Ⅎ'wnnf 34905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-bj-nnf 34906

This theorem is referenced by: (None)