![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > csbiegf | GIF version |
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
csbiegf.1 | ⊢ (𝐴 ∈ 𝑉 → Ⅎ𝑥𝐶) |
csbiegf.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
csbiegf | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbiegf.2 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
2 | 1 | ax-gen 1426 | . 2 ⊢ ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) |
3 | csbiegf.1 | . . 3 ⊢ (𝐴 ∈ 𝑉 → Ⅎ𝑥𝐶) | |
4 | csbiebt 3044 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | |
5 | 3, 4 | mpdan 418 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
6 | 2, 5 | mpbii 147 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1330 = wceq 1332 ∈ wcel 1481 Ⅎwnfc 2269 ⦋csb 3007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-sbc 2914 df-csb 3008 |
This theorem is referenced by: csbief 3049 sbcco3g 3062 csbco3g 3063 fmptcof 5595 fmpoco 6121 iseqf1olemjpcl 10299 iseqf1olemqpcl 10300 iseqf1olemfvp 10301 seq3f1olemqsum 10304 sumsnf 11210 |
Copyright terms: Public domain | W3C validator |