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Mirrors > Home > MPE Home > Th. List > vtocl3ga | Structured version Visualization version GIF version |
Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.) |
Ref | Expression |
---|---|
vtocl3ga.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtocl3ga.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
vtocl3ga.3 | ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) |
vtocl3ga.4 | ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑) |
Ref | Expression |
---|---|
vtocl3ga | ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2764 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2764 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2764 | . 2 ⊢ Ⅎ𝑧𝐴 | |
4 | nfcv 2764 | . 2 ⊢ Ⅎ𝑦𝐵 | |
5 | nfcv 2764 | . 2 ⊢ Ⅎ𝑧𝐵 | |
6 | nfcv 2764 | . 2 ⊢ Ⅎ𝑧𝐶 | |
7 | nfv 1843 | . 2 ⊢ Ⅎ𝑥𝜓 | |
8 | nfv 1843 | . 2 ⊢ Ⅎ𝑦𝜒 | |
9 | nfv 1843 | . 2 ⊢ Ⅎ𝑧𝜃 | |
10 | vtocl3ga.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
11 | vtocl3ga.2 | . 2 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
12 | vtocl3ga.3 | . 2 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) | |
13 | vtocl3ga.4 | . 2 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑) | |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | vtocl3gaf 3275 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 |
This theorem is referenced by: preq12bg 4386 prel12g 4387 pocl 5042 jensenlem2 24714 |
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