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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 329 | . . 3 ⊢ ([〈𝑧, 𝑤〉]𝑅 = 𝐵 → ((𝐴 ∈ 𝑆 → 𝜓) ↔ (𝐴 ∈ 𝑆 → 𝜒))) |
4 | 2ecoptocl.2 | . . . . . 6 ⊢ ([〈𝑥, 𝑦〉]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | imbi2d 329 | . . . . 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 397 | . . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜑)) |
8 | 1, 5, 7 | ecoptocl 7989 | . . . 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 7989 | . 2 ⊢ (𝐵 ∈ 𝑆 → (𝐴 ∈ 𝑆 → 𝜒)) |
11 | 10 | impcom 394 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 〈cop 4322 × cxp 5247 [cec 7894 / cqs 7895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-xp 5255 df-cnv 5257 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-ec 7898 df-qs 7902 |
This theorem is referenced by: 3ecoptocl 7991 ecovcom 8006 addclsr 10106 mulclsr 10107 ltsosr 10117 mulgt0sr 10128 |
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