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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 3492 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | vtoclgf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | issetf 3488 | . . 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 232 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝜓) |
8 | 4, 7 | exlimi 2210 | . . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓) |
9 | 3, 8 | sylbi 216 | . 2 ⊢ (𝐴 ∈ V → 𝜓) |
10 | 1, 9 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∃wex 1781 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 Vcvv 3474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-v 3476 |
This theorem is referenced by: vtocl2gf 3560 vtocl3gf 3561 vtoclgaf 3564 elabgf 3664 fsumsplit1 15695 ssiun2sf 32046 subtr 35502 subtr2 35503 supxrgere 44342 supxrgelem 44346 supxrge 44347 fmuldfeqlem1 44597 fprodcnlem 44614 climsuse 44623 dvnmptdivc 44953 dvmptfprodlem 44959 stoweidlem59 45074 fourierdlem31 45153 sge0f1o 45397 sge0fodjrnlem 45431 |
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