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| Mirrors > Home > MPE Home > Th. List > csbiedf | Structured version Visualization version 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 412 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → 𝐵 = 𝐶)) | 
| 4 | 1, 3 | alrimi 2212 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)) | 
| 5 | csbiedf.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 6 | csbiedf.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐶) | |
| 7 | csbiebt 3927 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | |
| 8 | 5, 6, 7 | syl2anc 584 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | 
| 9 | 4, 8 | mpbid 232 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1537 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2889 ⦋csb 3898 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-v 3481 df-sbc 3788 df-csb 3899 | 
| This theorem is referenced by: csbie2t 3936 fvmptdf 7021 fsumsplit1 15782 fprodsplit1f 16027 natpropd 18025 fucpropd 18026 gsummptf1o 19982 gsummpt2d 33053 gsummptfsf1o 33058 mnringvald 44232 sumsnd 45036 | 
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