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Mirrors > Home > ILE Home > Th. List > csbied2 | GIF version |
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
csbied2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
csbied2.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
csbied2.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
csbied2 | ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbied2.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | id 19 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
3 | csbied2.2 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 2, 3 | sylan9eqr 2172 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵) |
5 | csbied2.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷) | |
6 | 4, 5 | syldan 280 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷) |
7 | 1, 6 | csbied 3016 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 ⦋csb 2975 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-sbc 2883 df-csb 2976 |
This theorem is referenced by: (None) |
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