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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 2103). Version of sbied 2545 and sbiedv 2546 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 2102 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) | |
2 | sbiedvw.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
3 | 2 | expcom 416 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒))) |
4 | 3 | pm5.74d 275 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
5 | 4 | sbievw 2103 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) |
6 | 1, 5 | bitr3i 279 | . 2 ⊢ ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → 𝜒)) |
7 | 6 | pm5.74ri 274 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 |
This theorem is referenced by: 2sbievw 2105 iscatd2 16952 |
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