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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 2332 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2332 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2332 | . 2 ⊢ Ⅎ𝑦𝐵 | |
4 | nfv 1539 | . 2 ⊢ Ⅎ𝑥𝜓 | |
5 | nfv 1539 | . 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 2814 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 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: uniprg 3839 intprg 3892 opthg 4256 opelopabsb 4278 unexb 4460 vtoclr 4692 elimasng 5014 cnvsng 5132 funopg 5269 f1osng 5521 fsng 5710 fvsng 5733 op1stg 6176 op2ndg 6177 xpsneng 6849 xpcomeng 6855 mhmlem 13071 bdunexb 15150 bj-unexg 15151 |
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