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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfbd | Structured version Visualization version GIF version | ||
| 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 36726. (Contributed by BJ, 27-Aug-2023.) |
| Ref | Expression |
|---|---|
| bj-nnfbd.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| bj-nnfbd | ⊢ (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-nnfbd.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | alrimiv 1927 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
| 3 | bj-nnfbi 36726 | . 2 ⊢ (((𝜓 ↔ 𝜒) ∧ ∀𝑥(𝜓 ↔ 𝜒)) → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒)) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 Ⅎ'wnnf 36724 |
| 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-bj-nnf 36725 |
| This theorem is referenced by: (None) |
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