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Mirrors > Home > ILE Home > Th. List > csbiedf | GIF version |
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
csbiedf.1 | ⊢ Ⅎ𝑥𝜑 |
csbiedf.2 | ⊢ (𝜑 → Ⅎ𝑥𝐶) |
csbiedf.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
csbiedf.4 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
csbiedf | ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbiedf.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | csbiedf.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) | |
3 | 2 | ex 115 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → 𝐵 = 𝐶)) |
4 | 1, 3 | alrimi 1533 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)) |
5 | csbiedf.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | csbiedf.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐶) | |
7 | csbiebt 3111 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | |
8 | 5, 6, 7 | syl2anc 411 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
9 | 4, 8 | mpbid 147 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1362 = wceq 1364 Ⅎwnf 1471 ∈ wcel 2160 Ⅎwnfc 2319 ⦋csb 3072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-sbc 2978 df-csb 3073 |
This theorem is referenced by: csbied 3118 csbie2t 3120 fprodsplit1f 11677 |
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