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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 2920 | . . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ 𝐶 |
4 | vtocl2gaf.2 | . . . 4 ⊢ Ⅎ𝑦𝜒 | |
5 | 3, 4 | nfim 1894 | . . 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 1912 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐷 | |
10 | vtocl2gaf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
11 | 9, 10 | nfim 1894 | . . . . 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 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-v 3480 |
This theorem is referenced by: vtocl3gaf 3581 ovmpos 7581 ov2gf 7582 ov3 7596 pwfseqlem2 10697 cnmptcom 23702 |
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