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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-hbsb3t | Structured version Visualization version GIF version | ||
| Description: A theorem close to a closed form of hbsb3 2492. (Contributed by BJ, 2-May-2019.) |
| Ref | Expression |
|---|---|
| bj-hbsb3t | ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | spsbim 2072 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)) | |
| 2 | hbsb2a 2489 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | |
| 3 | 1, 2 | syl6 35 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 [wsb 2064 |
| 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 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 |
| This theorem is referenced by: bj-hbsb3 36790 bj-nfs1t 36791 |
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