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| Mirrors > Home > MPE Home > Th. List > vtoclgf | Structured version Visualization version GIF version | ||
| Description: Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of disjoint variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.) | 
| Ref | Expression | 
|---|---|
| vtoclgf.1 | ⊢ Ⅎ𝑥𝐴 | 
| vtoclgf.2 | ⊢ Ⅎ𝑥𝜓 | 
| vtoclgf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| vtoclgf.4 | ⊢ 𝜑 | 
| Ref | Expression | 
|---|---|
| vtoclgf | ⊢ (𝐴 ∈ 𝑉 → 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elex 3500 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
| 2 | vtoclgf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 2 | issetf 3496 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | 
| 4 | vtoclgf.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 5 | vtoclgf.4 | . . . . 5 ⊢ 𝜑 | |
| 6 | vtoclgf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 7 | 5, 6 | mpbii 233 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝜓) | 
| 8 | 4, 7 | exlimi 2216 | . . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓) | 
| 9 | 3, 8 | sylbi 217 | . 2 ⊢ (𝐴 ∈ V → 𝜓) | 
| 10 | 1, 9 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∃wex 1778 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2889 Vcvv 3479 | 
| 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-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 | 
| This theorem is referenced by: vtocl2gf 3571 vtocl3gf 3572 vtoclgaf 3575 elabgf 3673 fsumsplit1 15782 ssiun2sf 32573 subtr 36316 subtr2 36317 supxrgere 45349 supxrgelem 45353 supxrge 45354 fmuldfeqlem1 45602 climsuse 45628 dvnmptdivc 45958 dvmptfprodlem 45964 stoweidlem59 46079 fourierdlem31 46158 sge0fodjrnlem 46436 | 
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