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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 3499 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | vtoclgf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | issetf 3495 | . . 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 2215 | . . 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 1537 ∃wex 1776 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 Vcvv 3478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-v 3480 |
This theorem is referenced by: vtocl2gf 3572 vtocl3gf 3573 vtoclgaf 3576 elabgf 3675 fsumsplit1 15778 ssiun2sf 32580 subtr 36297 subtr2 36298 supxrgere 45283 supxrgelem 45287 supxrge 45288 fmuldfeqlem1 45538 climsuse 45564 dvnmptdivc 45894 dvmptfprodlem 45900 stoweidlem59 46015 fourierdlem31 46094 sge0fodjrnlem 46372 |
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