![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 2229 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2229 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2229 | . 2 ⊢ Ⅎ𝑦𝐵 | |
4 | nfv 1467 | . 2 ⊢ Ⅎ𝑥𝜓 | |
5 | nfv 1467 | . 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 2682 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1290 ∈ wcel 1439 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 |
This theorem is referenced by: uniprg 3674 intprg 3727 opthg 4074 opelopabsb 4096 unexb 4277 vtoclr 4499 elimasng 4813 cnvsng 4929 funopg 5061 f1osng 5307 fsng 5484 fvsng 5507 op1stg 5935 op2ndg 5936 xpsneng 6592 xpcomeng 6598 bdunexb 12084 bj-unexg 12085 |
Copyright terms: Public domain | W3C validator |