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Mirrors > Home > ILE Home > Th. List > eqoprab2b | GIF version |
Description: Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4063. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
eqoprab2b | ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssoprab2b 5614 | . . 3 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓)) | |
2 | ssoprab2b 5614 | . . 3 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑)) | |
3 | 1, 2 | anbi12i 448 | . 2 ⊢ (({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ∧ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) ↔ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑))) |
4 | eqss 3024 | . 2 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ∧ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑})) | |
5 | 2albiim 1418 | . . . 4 ⊢ (∀𝑦∀𝑧(𝜑 ↔ 𝜓) ↔ (∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑦∀𝑧(𝜓 → 𝜑))) | |
6 | 5 | albii 1400 | . . 3 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) ↔ ∀𝑥(∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑦∀𝑧(𝜓 → 𝜑))) |
7 | 19.26 1411 | . . 3 ⊢ (∀𝑥(∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑦∀𝑧(𝜓 → 𝜑)) ↔ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑))) | |
8 | 6, 7 | bitri 182 | . 2 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑))) |
9 | 3, 4, 8 | 3bitr4i 210 | 1 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1283 = wceq 1285 ⊆ wss 2983 {coprab 5565 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-pow 3969 ax-pr 3993 ax-setind 4309 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-v 2612 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-oprab 5568 |
This theorem is referenced by: mpt22eqb 5662 |
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