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| Mirrors > Home > MPE Home > Th. List > opabresex2d | Structured version Visualization version GIF version | ||
| Description: Obsolete version of opabresex2 7485 as of 13-Dec-2024. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| opabresex2d.1 | ⊢ ((𝜑 ∧ 𝑥(𝑊‘𝐺)𝑦) → 𝜓) |
| opabresex2d.2 | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ∈ 𝑉) |
| Ref | Expression |
|---|---|
| opabresex2d | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opabresex2d.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑊‘𝐺)𝑦) → 𝜓) | |
| 2 | 1 | ex 412 | . . 3 ⊢ (𝜑 → (𝑥(𝑊‘𝐺)𝑦 → 𝜓)) |
| 3 | 2 | alrimivv 1928 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑊‘𝐺)𝑦 → 𝜓)) |
| 4 | opabresex2d.2 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ∈ 𝑉) | |
| 5 | opabbrex 7484 | . 2 ⊢ ((∀𝑥∀𝑦(𝑥(𝑊‘𝐺)𝑦 → 𝜓) ∧ {〈𝑥, 𝑦〉 ∣ 𝜓} ∈ 𝑉) → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V) | |
| 6 | 3, 4, 5 | syl2anc 584 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 {copab 5205 ‘cfv 6561 |
| 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-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-opab 5206 |
| This theorem is referenced by: mptmpoopabbrdOLDOLD 8108 |
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