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Mirrors > Home > MPE Home > Th. List > 2ecoptocl | Structured version Visualization version GIF version |
Description: Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.) |
Ref | Expression |
---|---|
2ecoptocl.1 | ⊢ 𝑆 = ((𝐶 × 𝐷) / 𝑅) |
2ecoptocl.2 | ⊢ ([⟨𝑥, 𝑦⟩]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) |
2ecoptocl.3 | ⊢ ([⟨𝑧, 𝑤⟩]𝑅 = 𝐵 → (𝜓 ↔ 𝜒)) |
2ecoptocl.4 | ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) → 𝜑) |
Ref | Expression |
---|---|
2ecoptocl | ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2ecoptocl.1 | . . 3 ⊢ 𝑆 = ((𝐶 × 𝐷) / 𝑅) | |
2 | 2ecoptocl.3 | . . . 4 ⊢ ([⟨𝑧, 𝑤⟩]𝑅 = 𝐵 → (𝜓 ↔ 𝜒)) | |
3 | 2 | imbi2d 341 | . . 3 ⊢ ([⟨𝑧, 𝑤⟩]𝑅 = 𝐵 → ((𝐴 ∈ 𝑆 → 𝜓) ↔ (𝐴 ∈ 𝑆 → 𝜒))) |
4 | 2ecoptocl.2 | . . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | imbi2d 341 | . . . . 5 ⊢ ([⟨𝑥, 𝑦⟩]𝑅 = 𝐴 → (((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜑) ↔ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜓))) |
6 | 2ecoptocl.4 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) → 𝜑) | |
7 | 6 | ex 414 | . . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜑)) |
8 | 1, 5, 7 | ecoptocl 8747 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜓)) |
9 | 8 | com12 32 | . . 3 ⊢ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → (𝐴 ∈ 𝑆 → 𝜓)) |
10 | 1, 3, 9 | ecoptocl 8747 | . 2 ⊢ (𝐵 ∈ 𝑆 → (𝐴 ∈ 𝑆 → 𝜒)) |
11 | 10 | impcom 409 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⟨cop 4593 × cxp 5632 [cec 8647 / cqs 8648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ec 8651 df-qs 8655 |
This theorem is referenced by: 3ecoptocl 8749 ecovcom 8763 addclsr 11020 mulclsr 11021 ltsosr 11031 mulgt0sr 11042 |
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