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Mirrors > Home > ILE Home > Th. List > vtocl2ga | GIF version |
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.) |
Ref | Expression |
---|---|
vtocl2ga.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtocl2ga.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
vtocl2ga.3 | ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) |
Ref | Expression |
---|---|
vtocl2ga | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2332 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2332 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2332 | . 2 ⊢ Ⅎ𝑦𝐵 | |
4 | nfv 1539 | . 2 ⊢ Ⅎ𝑥𝜓 | |
5 | nfv 1539 | . 2 ⊢ Ⅎ𝑦𝜒 | |
6 | vtocl2ga.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | vtocl2ga.2 | . 2 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
8 | vtocl2ga.3 | . 2 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | vtocl2gaf 2819 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 |
This theorem is referenced by: caovcan 6056 genipv 7526 fsumrelem 11497 |
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