Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3185 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | vtoclgf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | issetf 3181 | . . 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 222 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝜓) |
8 | 4, 7 | exlimi 2073 | . . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓) |
9 | 3, 8 | sylbi 206 | . 2 ⊢ (𝐴 ∈ V → 𝜓) |
10 | 1, 9 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∃wex 1695 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 |
This theorem is referenced by: vtocl2gf 3241 vtocl3gf 3242 vtoclgaf 3244 elabgf 3317 fprodsplit1f 14560 ssiun2sf 28760 subtr 31478 subtr2 31479 dmrelrnrel 38414 supxrgere 38490 supxrgelem 38494 supxrge 38495 fsumsplit1 38639 fmuldfeqlem1 38649 fprodcnlem 38666 climsuse 38675 dvnmptdivc 38828 dvmptfprodlem 38834 stoweidlem59 38952 fourierdlem31 39031 sge0f1o 39275 sge0fodjrnlem 39309 salpreimagelt 39595 salpreimalegt 39597 |
Copyright terms: Public domain | W3C validator |