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Mirrors > Home > MPE Home > Th. List > vtocl4g | Structured version Visualization version GIF version |
Description: Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.) |
Ref | Expression |
---|---|
vtocl4ga.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtocl4ga.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
vtocl4ga.3 | ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜌)) |
vtocl4ga.4 | ⊢ (𝑤 = 𝐷 → (𝜌 ↔ 𝜃)) |
vtocl4g.5 | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtocl4g | ⊢ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtocl4ga.3 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜌)) | |
2 | 1 | imbi2d 330 | . . 3 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜒) ↔ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜌))) |
3 | vtocl4ga.4 | . . . 4 ⊢ (𝑤 = 𝐷 → (𝜌 ↔ 𝜃)) | |
4 | 3 | imbi2d 330 | . . 3 ⊢ (𝑤 = 𝐷 → (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜌) ↔ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜃))) |
5 | vtocl4ga.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | vtocl4ga.2 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
7 | vtocl4g.5 | . . . 4 ⊢ 𝜑 | |
8 | 5, 6, 7 | vtocl2g 3270 | . . 3 ⊢ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜒) |
9 | 2, 4, 8 | vtocl2g 3270 | . 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇) → ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜃)) |
10 | 9 | impcom 446 | 1 ⊢ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = 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-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: vtocl4ga 3278 |
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