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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-hbs1 | Structured version Visualization version GIF version |
Description: Version of hbsb2 2484 with a disjoint variable condition, which does not require ax-13 2370, and removal of ax-13 2370 from hbs1 2264. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-hbs1 | ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6 2086 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | 1 | biimpri 227 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) |
3 | 2 | axc4i 2314 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥[𝑦 / 𝑥]𝜑) |
4 | 1, 3 | sylbi 216 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 [wsb 2065 |
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 1911 ax-6 1969 ax-7 2009 ax-10 2135 ax-12 2169 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-ex 1780 df-nf 1784 df-sb 2066 |
This theorem is referenced by: bj-nfs1v 35036 |
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