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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sbft | Structured version Visualization version GIF version |
Description: Version of sbft 2270 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.) |
Ref | Expression |
---|---|
bj-sbft | ⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbe 2088 | . . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑) | |
2 | bj-nnfe 34062 | . . 3 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) | |
3 | 1, 2 | syl5 34 | . 2 ⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 → 𝜑)) |
4 | bj-nnfa 34060 | . . 3 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | |
5 | stdpc4 2073 | . . 3 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑) | |
6 | 4, 5 | syl6 35 | . 2 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → [𝑡 / 𝑥]𝜑)) |
7 | 3, 6 | impbid 214 | 1 ⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∃wex 1780 [wsb 2069 Ⅎ'wnnf 34055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-bj-nnf 34056 |
This theorem is referenced by: (None) |
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