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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 2175 | . 2 ⊢ ℲxA | |
2 | nfcv 2175 | . 2 ⊢ ℲyA | |
3 | nfcv 2175 | . 2 ⊢ ℲyB | |
4 | nfv 1418 | . 2 ⊢ Ⅎxψ | |
5 | nfv 1418 | . 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 2609 | 1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → χ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 |
This theorem is referenced by: uniprg 3586 intprg 3639 opthg 3966 opelopabsb 3988 unexb 4143 vtoclr 4331 elimasng 4636 cnvsng 4749 funopg 4877 f1osng 5110 fsng 5279 fvsng 5302 op1stg 5719 op2ndg 5720 xpsneng 6232 xpcomeng 6238 bdunexb 9375 bj-unexg 9376 |
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