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| Mirrors > Home > MPE Home > Th. List > csbiegf | Structured version Visualization version GIF version | ||
| Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
| Ref | Expression |
|---|---|
| csbiegf.1 | ⊢ (𝐴 ∈ 𝑉 → Ⅎ𝑥𝐶) |
| csbiegf.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| csbiegf | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | csbiegf.2 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
| 2 | 1 | ax-gen 1795 | . 2 ⊢ ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) |
| 3 | csbiegf.1 | . . 3 ⊢ (𝐴 ∈ 𝑉 → Ⅎ𝑥𝐶) | |
| 4 | csbiebt 3928 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | |
| 5 | 3, 4 | mpdan 687 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
| 6 | 2, 5 | mpbii 233 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ⦋csb 3899 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 df-sbc 3789 df-csb 3900 |
| This theorem is referenced by: csbief 3933 sbcco3gw 4425 sbcco3g 4430 csbco3g 4431 fmptcof 7150 fmpoco 8120 sumsnf 15779 prodsn 15998 prodsnf 16000 bpolylem 16084 pcmpt 16930 chfacfpmmulfsupp 22869 elmptrab 23835 dvfsumrlim3 26074 itgsubstlem 26089 itgsubst 26090 ifeqeqx 32555 disjunsn 32607 sbcaltop 35982 unirep 37721 cdleme31so 40381 cdleme31sn 40382 cdleme31sn1 40383 cdleme31se 40384 cdleme31se2 40385 cdleme31sc 40386 cdleme31sde 40387 cdleme31sn2 40391 cdlemeg47rv2 40512 cdlemk41 40922 monotuz 42953 oddcomabszz 42956 |
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