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Mirrors > Home > MPE Home > Th. List > oprabbid | Structured version Visualization version GIF version |
Description: Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.) |
Ref | Expression |
---|---|
oprabbid.1 | ⊢ Ⅎ𝑥𝜑 |
oprabbid.2 | ⊢ Ⅎ𝑦𝜑 |
oprabbid.3 | ⊢ Ⅎ𝑧𝜑 |
oprabbid.4 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
oprabbid | ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oprabbid.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | oprabbid.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
3 | oprabbid.3 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
4 | oprabbid.4 | . . . . . . 7 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
5 | 4 | anbi2d 628 | . . . . . 6 ⊢ (𝜑 → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒))) |
6 | 3, 5 | exbid 2208 | . . . . 5 ⊢ (𝜑 → (∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒))) |
7 | 2, 6 | exbid 2208 | . . . 4 ⊢ (𝜑 → (∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒))) |
8 | 1, 7 | exbid 2208 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒))) |
9 | 8 | abbidv 2795 | . 2 ⊢ (𝜑 → {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)}) |
10 | df-oprab 7409 | . 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} | |
11 | df-oprab 7409 | . 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)} | |
12 | 9, 10, 11 | 3eqtr4g 2791 | 1 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∃wex 1773 Ⅎwnf 1777 {cab 2703 ⟨cop 4629 {coprab 7406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-oprab 7409 |
This theorem is referenced by: mpoeq123 7477 |
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