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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 2235 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2235 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2235 | . 2 ⊢ Ⅎ𝑦𝐵 | |
4 | nfv 1473 | . 2 ⊢ Ⅎ𝑥𝜓 | |
5 | nfv 1473 | . 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 2700 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1296 ∈ wcel 1445 |
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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 |
This theorem is referenced by: caovcan 5847 genipv 7165 fsumrelem 11029 |
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