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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 2304 and sbievw 2088. Usage of this theorem is discouraged because it depends on ax-13 2367. (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 2486 | . . 3 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦 | |
2 | sbie.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | sbimi 2070 | . . 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 2258 | . . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓) |
7 | 6 | sblbis 2299 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
8 | 4, 7 | mpbi 229 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 Ⅎwnf 1778 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-nf 1779 df-sb 2061 |
This theorem is referenced by: sbied 2498 2sbiev 2500 cbvmo 2595 cbveu 2599 cbvab 2804 clelsb2OLD 2858 cbvralf 3352 cbvreu 3420 cbvrab 3469 nfcdeq 3771 cbvralcsf 3935 cbvreucsf 3937 cbvrabcsf 3938 cbvopab1g 5218 cbvmptfg 5252 cbviota 6504 cbvriota 7384 nd1 10604 nd2 10605 sbcrexgOLD 42199 |
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