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Mirrors > Home > MPE Home > Th. List > vtocl2gaf | Structured version Visualization version GIF version |
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.) |
Ref | Expression |
---|---|
vtocl2gaf.a | ⊢ Ⅎ𝑥𝐴 |
vtocl2gaf.b | ⊢ Ⅎ𝑦𝐴 |
vtocl2gaf.c | ⊢ Ⅎ𝑦𝐵 |
vtocl2gaf.1 | ⊢ Ⅎ𝑥𝜓 |
vtocl2gaf.2 | ⊢ Ⅎ𝑦𝜒 |
vtocl2gaf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtocl2gaf.4 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
vtocl2gaf.5 | ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) |
Ref | Expression |
---|---|
vtocl2gaf | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtocl2gaf.a | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | vtocl2gaf.b | . . 3 ⊢ Ⅎ𝑦𝐴 | |
3 | vtocl2gaf.c | . . 3 ⊢ Ⅎ𝑦𝐵 | |
4 | 1 | nfel1 2916 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐶 |
5 | nfv 1922 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐷 | |
6 | 4, 5 | nfan 1907 | . . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) |
7 | vtocl2gaf.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
8 | 6, 7 | nfim 1904 | . . 3 ⊢ Ⅎ𝑥((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜓) |
9 | 2 | nfel1 2916 | . . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ 𝐶 |
10 | 3 | nfel1 2916 | . . . . 5 ⊢ Ⅎ𝑦 𝐵 ∈ 𝐷 |
11 | 9, 10 | nfan 1907 | . . . 4 ⊢ Ⅎ𝑦(𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) |
12 | vtocl2gaf.2 | . . . 4 ⊢ Ⅎ𝑦𝜒 | |
13 | 11, 12 | nfim 1904 | . . 3 ⊢ Ⅎ𝑦((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) |
14 | eleq1 2821 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
15 | 14 | anbi1d 633 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
16 | vtocl2gaf.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
17 | 15, 16 | imbi12d 348 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) ↔ ((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜓))) |
18 | eleq1 2821 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷)) | |
19 | 18 | anbi2d 632 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))) |
20 | vtocl2gaf.4 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
21 | 19, 20 | imbi12d 348 | . . 3 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜓) ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒))) |
22 | vtocl2gaf.5 | . . 3 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) | |
23 | 1, 2, 3, 8, 13, 17, 21, 22 | vtocl2gf 3477 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)) |
24 | 23 | pm2.43i 52 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-v 3403 |
This theorem is referenced by: ovmpos 7346 ov2gf 7347 ov3 7360 pwfseqlem2 10256 cnmptcom 22547 |
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