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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 2506) Usage of this theorem is discouraged because it depends on ax-13 2371. See sbiedw 2316, sbiedvw 2103 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 2307 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
3 | sbied.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
4 | 1, 3 | nfim1 2200 | . . . 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 2506 | . . 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 1790 [wsb 2073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-10 2144 ax-12 2178 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 df-sb 2074 |
This theorem is referenced by: sbiedv 2508 sbco2 2515 wl-equsb3 35323 |
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