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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 1926 | . 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 1535 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 {copab 5210 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-in 3970 df-ss 3980 df-opab 5211 |
This theorem is referenced by: mptmpoopabbrdOLDOLD 8107 |
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