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Mirrors > Home > MPE Home > Th. List > sbiedvw | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbievw 2093). Version of sbied 2500 and sbiedv 2501 with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Gino Giotto, 29-Jan-2024.) |
Ref | Expression |
---|---|
sbiedvw.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
sbiedvw | ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbrimvw 2092 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) | |
2 | sbiedvw.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
3 | 2 | expcom 412 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒))) |
4 | 3 | pm5.74d 272 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
5 | 4 | sbievw 2093 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) |
6 | 1, 5 | bitr3i 276 | . 2 ⊢ ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → 𝜒)) |
7 | 6 | pm5.74ri 271 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 [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 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1780 df-sb 2066 |
This theorem is referenced by: 2sbievw 2095 iscatd2 17629 bj-elabd2ALT 36108 |
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