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Mirrors > Home > MPE Home > Th. List > vtocl3gaOLD | Structured version Visualization version GIF version |
Description: Obsolete version of vtocl3ga 3565 as of 3-Oct-2024. (Contributed by NM, 20-Aug-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vtocl3gaOLD.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtocl3gaOLD.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
vtocl3gaOLD.3 | ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) |
vtocl3gaOLD.4 | ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑) |
Ref | Expression |
---|---|
vtocl3gaOLD | ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2898 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2898 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2898 | . 2 ⊢ Ⅎ𝑧𝐴 | |
4 | nfcv 2898 | . 2 ⊢ Ⅎ𝑦𝐵 | |
5 | nfcv 2898 | . 2 ⊢ Ⅎ𝑧𝐵 | |
6 | nfcv 2898 | . 2 ⊢ Ⅎ𝑧𝐶 | |
7 | nfv 1910 | . 2 ⊢ Ⅎ𝑥𝜓 | |
8 | nfv 1910 | . 2 ⊢ Ⅎ𝑦𝜒 | |
9 | nfv 1910 | . 2 ⊢ Ⅎ𝑧𝜃 | |
10 | vtocl3gaOLD.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
11 | vtocl3gaOLD.2 | . 2 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
12 | vtocl3gaOLD.3 | . 2 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) | |
13 | vtocl3gaOLD.4 | . 2 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑) | |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | vtocl3gaf 3564 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-v 3471 |
This theorem is referenced by: (None) |
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