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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfbit | Structured version Visualization version GIF version |
Description: Nonfreeness in both sides implies nonfreeness in the biconditional. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnfbit | ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nnfim 35240 | . . 3 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 → 𝜓)) | |
2 | bj-nnfim 35240 | . . . 4 ⊢ ((Ⅎ'𝑥𝜓 ∧ Ⅎ'𝑥𝜑) → Ⅎ'𝑥(𝜓 → 𝜑)) | |
3 | 2 | ancoms 460 | . . 3 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜓 → 𝜑)) |
4 | bj-nnfan 35242 | . . 3 ⊢ ((Ⅎ'𝑥(𝜑 → 𝜓) ∧ Ⅎ'𝑥(𝜓 → 𝜑)) → Ⅎ'𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) | |
5 | 1, 3, 4 | syl2anc 585 | . 2 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) |
6 | dfbi2 476 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) | |
7 | 6 | bicomi 223 | . . 3 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ (𝜑 ↔ 𝜓)) |
8 | 7 | bj-nnfbii 35221 | . 2 ⊢ (Ⅎ'𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ Ⅎ'𝑥(𝜑 ↔ 𝜓)) |
9 | 5, 8 | sylib 217 | 1 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 Ⅎ'wnnf 35217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-bj-nnf 35218 |
This theorem is referenced by: (None) |
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