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Mirrors > Home > MPE Home > Th. List > sbied | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 2522) Usage of this theorem is discouraged because it depends on ax-13 2380. See sbiedw 2324, sbiedvw 2102 for variants using disjoint variables, but requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbied.1 | ⊢ Ⅎ𝑥𝜑 |
sbied.2 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
sbied.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
sbied | ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbied.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | sbrim 2310 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
3 | sbied.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
4 | 1, 3 | nfim1 2198 | . . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜒) |
5 | sbied.3 | . . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
6 | 5 | com12 32 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒))) |
7 | 6 | pm5.74d 276 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
8 | 4, 7 | sbie 2522 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) |
9 | 2, 8 | bitr3i 280 | . 2 ⊢ ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → 𝜒)) |
10 | 9 | pm5.74ri 275 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 Ⅎwnf 1786 [wsb 2070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-10 2143 ax-12 2176 ax-13 2380 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-ex 1783 df-nf 1787 df-sb 2071 |
This theorem is referenced by: sbiedv 2524 sbco2 2531 wl-equsb3 35222 |
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