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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.) (Proof shortened by Wolf Lammen, 31-May-2025.) | 
| 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.c | . . 3 ⊢ Ⅎ𝑦𝐵 | |
| 2 | vtocl2gaf.b | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
| 3 | 2 | nfel1 2922 | . . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ 𝐶 | 
| 4 | vtocl2gaf.2 | . . . 4 ⊢ Ⅎ𝑦𝜒 | |
| 5 | 3, 4 | nfim 1896 | . . 3 ⊢ Ⅎ𝑦(𝐴 ∈ 𝐶 → 𝜒) | 
| 6 | vtocl2gaf.4 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
| 7 | 6 | imbi2d 340 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 → 𝜓) ↔ (𝐴 ∈ 𝐶 → 𝜒))) | 
| 8 | vtocl2gaf.a | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
| 9 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐷 | |
| 10 | vtocl2gaf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
| 11 | 9, 10 | nfim 1896 | . . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐷 → 𝜓) | 
| 12 | vtocl2gaf.3 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 13 | 12 | imbi2d 340 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 ∈ 𝐷 → 𝜑) ↔ (𝑦 ∈ 𝐷 → 𝜓))) | 
| 14 | vtocl2gaf.5 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) | |
| 15 | 14 | ex 412 | . . . . 5 ⊢ (𝑥 ∈ 𝐶 → (𝑦 ∈ 𝐷 → 𝜑)) | 
| 16 | 8, 11, 13, 15 | vtoclgaf 3576 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝑦 ∈ 𝐷 → 𝜓)) | 
| 17 | 16 | com12 32 | . . 3 ⊢ (𝑦 ∈ 𝐷 → (𝐴 ∈ 𝐶 → 𝜓)) | 
| 18 | 1, 5, 7, 17 | vtoclgaf 3576 | . 2 ⊢ (𝐵 ∈ 𝐷 → (𝐴 ∈ 𝐶 → 𝜒)) | 
| 19 | 18 | impcom 407 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 | 
| This theorem is referenced by: vtocl3gaf 3581 ovmpos 7581 ov2gf 7582 ov3 7596 pwfseqlem2 10699 cnmptcom 23686 | 
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