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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 2481. (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 2068 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)) | |
2 | hbsb2a 2478 | . 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 1532 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-12 2167 ax-13 2366 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ex 1775 df-nf 1779 df-sb 2061 |
This theorem is referenced by: bj-hbsb3 36496 bj-nfs1t 36497 |
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