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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 2997 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐶 |
5 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐷 | |
6 | 4, 5 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) |
7 | vtocl2gaf.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
8 | 6, 7 | nfim 1896 | . . 3 ⊢ Ⅎ𝑥((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜓) |
9 | 2 | nfel1 2997 | . . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ 𝐶 |
10 | 3 | nfel1 2997 | . . . . 5 ⊢ Ⅎ𝑦 𝐵 ∈ 𝐷 |
11 | 9, 10 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑦(𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) |
12 | vtocl2gaf.2 | . . . 4 ⊢ Ⅎ𝑦𝜒 | |
13 | 11, 12 | nfim 1896 | . . 3 ⊢ Ⅎ𝑦((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) |
14 | eleq1 2903 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
15 | 14 | anbi1d 631 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
16 | vtocl2gaf.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
17 | 15, 16 | imbi12d 347 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) ↔ ((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜓))) |
18 | eleq1 2903 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷)) | |
19 | 18 | anbi2d 630 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))) |
20 | vtocl2gaf.4 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
21 | 19, 20 | imbi12d 347 | . . 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 3573 | . 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 208 ∧ wa 398 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 Ⅎwnfc 2964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 |
This theorem is referenced by: ovmpos 7301 ov2gf 7302 ov3 7314 pwfseqlem2 10084 cnmptcom 22289 |
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