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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 114 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → 𝐵 = 𝐶)) |
4 | 1, 3 | alrimi 1515 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)) |
5 | csbiedf.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | csbiedf.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐶) | |
7 | csbiebt 3088 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | |
8 | 5, 6, 7 | syl2anc 409 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
9 | 4, 8 | mpbid 146 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1346 = wceq 1348 Ⅎwnf 1453 ∈ wcel 2141 Ⅎwnfc 2299 ⦋csb 3049 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-sbc 2956 df-csb 3050 |
This theorem is referenced by: csbied 3095 csbie2t 3097 fprodsplit1f 11590 |
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