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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sbidmOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of sbidm 2545 temporarily kept here to check it gives no additional insight. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-sbidmOLD | ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsb2 2524 | . . 3 ⊢ [𝑦 / 𝑥]𝑦 = 𝑥 | |
2 | sbequ12r 2244 | . . . 4 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
3 | 2 | sbimi 2070 | . . 3 ⊢ ([𝑦 / 𝑥]𝑦 = 𝑥 → [𝑦 / 𝑥]([𝑦 / 𝑥]𝜑 ↔ 𝜑)) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ [𝑦 / 𝑥]([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
5 | sbbi 2308 | . 2 ⊢ ([𝑦 / 𝑥]([𝑦 / 𝑥]𝜑 ↔ 𝜑) ↔ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
6 | 4, 5 | mpbi 231 | 1 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 |
This theorem is referenced by: (None) |
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