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| Mirrors > Home > MPE Home > Th. List > eqopab2b | Structured version Visualization version GIF version | ||
| Description: Equivalence of ordered pair abstraction equality and biconditional. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker eqopab2bw 5553 when possible. (Contributed by Mario Carneiro, 4-Jan-2017.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| eqopab2b | ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssopab2b 5554 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) | |
| 2 | ssopab2b 5554 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∀𝑥∀𝑦(𝜓 → 𝜑)) | |
| 3 | 1, 2 | anbi12i 628 | . 2 ⊢ (({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} ∧ {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜑}) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑))) |
| 4 | eqss 3999 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} ∧ {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
| 5 | 2albiim 1890 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑))) | |
| 6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ⊆ wss 3951 {copab 5205 |
| 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-13 2377 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-nfc 2892 df-ral 3062 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 df-opab 5206 |
| This theorem is referenced by: opabbi 38172 |
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