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Mirrors > Home > MPE Home > Th. List > csbied2 | Structured version Visualization version 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 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
3 | csbied2.2 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 2, 3 | sylan9eqr 2796 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵) |
5 | csbied2.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷) | |
6 | 4, 5 | syldan 591 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷) |
7 | 1, 6 | csbied 3945 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ⦋csb 3907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-sbc 3791 df-csb 3908 |
This theorem is referenced by: prdsval 17501 cidfval 17720 monfval 17779 idfuval 17926 isnat 18001 fucco 18018 catcval 18153 xpcval 18232 1stfval 18246 2ndfval 18249 prfval 18254 evlf2 18274 curfval 18279 hofval 18308 ipoval 18587 mntoval 32956 mgcoval 32960 erlval 33244 rlocval 33245 poimirlem2 37608 rngcvalALTV 48108 ringcvalALTV 48132 |
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