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Mirrors > Home > MPE Home > Th. List > sbie | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit substitution. For versions requiring disjoint variables, but fewer axioms, see sbiev 2322 and sbievw 2100. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbie.1 | ⊢ Ⅎ𝑥𝜓 |
sbie.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbie | ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsb1 2509 | . . 3 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦 | |
2 | sbie.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | sbimi 2079 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥](𝜑 ↔ 𝜓)) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ [𝑦 / 𝑥](𝜑 ↔ 𝜓) |
5 | sbie.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
6 | 5 | sbf 2268 | . . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓) |
7 | 6 | sblbis 2314 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
8 | 4, 7 | mpbi 233 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 Ⅎwnf 1785 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 |
This theorem is referenced by: sbied 2522 2sbiev 2524 cbvmo 2665 cbveu 2668 cbvab 2869 cbvralf 3385 cbvreu 3394 cbvrab 3438 nfcdeq 3716 cbvralcsf 3870 cbvreucsf 3872 cbvrabcsf 3873 cbvopab1g 5104 cbvmptfg 5130 cbviota 6292 cbvriota 7106 nd1 9998 nd2 9999 sbcrexgOLD 39726 |
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