| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > copsex2dv | Structured version Visualization version GIF version | ||
| Description: Implicit substitution deduction for ordered pairs. (Contributed by Thierry Arnoux, 4-May-2025.) |
| Ref | Expression |
|---|---|
| copsex2dv.a | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
| copsex2dv.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
| copsex2dv.1 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| copsex2dv | ⊢ (𝜑 → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | copsex2dv.1 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | ex 412 | . . 3 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))) |
| 3 | 2 | alrimivv 1928 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))) |
| 4 | copsex2dv.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
| 5 | copsex2dv.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
| 6 | copsex2t 5497 | . 2 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ 𝜒)) | |
| 7 | 3, 4, 5, 6 | syl12anc 837 | 1 ⊢ (𝜑 → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 〈cop 4632 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 |
| This theorem is referenced by: brab2d 32619 brab2dd 48741 |
| Copyright terms: Public domain | W3C validator |