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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 2330 and sbievw 2103. Usage of this theorem is discouraged because it depends on ax-13 2390. (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 2530 | . . 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 2271 | . . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓) |
7 | 6 | sblbis 2319 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
8 | 4, 7 | mpbi 232 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 Ⅎwnf 1784 [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 ax-10 2145 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: sbied 2545 2sbiev 2547 cbvmo 2689 cbveu 2691 cbvab 2891 clelsb3fOLD 2983 cbvralf 3439 cbvreu 3447 cbvrab 3490 nfcdeq 3768 cbvralcsf 3925 cbvreucsf 3927 cbvrabcsf 3928 cbvopab1g 5140 cbvmptfg 5166 cbviota 6323 cbvriota 7127 nd1 10009 nd2 10010 sbcrexgOLD 39431 |
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