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Mirrors > Home > MPE Home > Th. List > copsex2ga | Structured version Visualization version GIF version |
Description: Implicit substitution inference for ordered pairs. Compare copsex2g 4987. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
copsex2ga.1 | ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
copsex2ga | ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpss 5159 | . . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V) | |
2 | 1 | sseli 3632 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V)) |
3 | copsex2ga.1 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) | |
4 | 3 | copsex2gb 5263 | . . 3 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑)) |
5 | 4 | baibr 965 | . 2 ⊢ (𝐴 ∈ (V × V) → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
6 | 2, 5 | syl 17 | 1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∃wex 1744 ∈ wcel 2030 Vcvv 3231 〈cop 4216 × cxp 5141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-opab 4746 df-xp 5149 |
This theorem is referenced by: (None) |
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