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Mirrors > Home > ILE Home > Th. List > vtocl2g | GIF version |
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.) |
Ref | Expression |
---|---|
vtocl2g.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtocl2g.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
vtocl2g.3 | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtocl2g | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2178 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2178 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2178 | . 2 ⊢ Ⅎ𝑦𝐵 | |
4 | nfv 1421 | . 2 ⊢ Ⅎ𝑥𝜓 | |
5 | nfv 1421 | . 2 ⊢ Ⅎ𝑦𝜒 | |
6 | vtocl2g.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | vtocl2g.2 | . 2 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
8 | vtocl2g.3 | . 2 ⊢ 𝜑 | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | vtocl2gf 2615 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 |
This theorem is referenced by: uniprg 3595 intprg 3648 opthg 3975 opelopabsb 3997 unexb 4177 vtoclr 4388 elimasng 4693 cnvsng 4806 funopg 4934 f1osng 5167 fsng 5336 fvsng 5359 op1stg 5777 op2ndg 5778 xpsneng 6296 xpcomeng 6302 bdunexb 10040 bj-unexg 10041 |
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