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Theorem bj-nnfbd 36708
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 36707. (Contributed by BJ, 27-Aug-2023.)
Hypothesis
Ref Expression
bj-nnfbd.1 (𝜑 → (𝜓𝜒))
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
StepHypRef Expression
1 bj-nnfbd.1 . 2 (𝜑 → (𝜓𝜒))
21alrimiv 1924 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
3 bj-nnfbi 36707 . 2 (((𝜓𝜒) ∧ ∀𝑥(𝜓𝜒)) → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
41, 2, 3syl2anc 584 1 (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1534  Ⅎ'wnnf 36705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-bj-nnf 36706
This theorem is referenced by: (None)
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