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Mirrors > Home > MPE Home > Th. List > eqoprab2b | Structured version Visualization version GIF version |
Description: Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 5545. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker eqoprab2bw 7463 when possible. (Contributed by Mario Carneiro, 4-Jan-2017.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eqoprab2b | ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssoprab2b 7462 | . . 3 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓)) | |
2 | ssoprab2b 7462 | . . 3 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑)) | |
3 | 1, 2 | anbi12i 627 | . 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 3993 | . 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 1893 | . . . 4 ⊢ (∀𝑦∀𝑧(𝜑 ↔ 𝜓) ↔ (∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑦∀𝑧(𝜓 → 𝜑))) | |
6 | 5 | albii 1821 | . . 3 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) ↔ ∀𝑥(∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑦∀𝑧(𝜓 → 𝜑))) |
7 | 19.26 1873 | . . 3 ⊢ (∀𝑥(∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑦∀𝑧(𝜓 → 𝜑)) ↔ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑))) | |
8 | 6, 7 | bitri 274 | . 2 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦∀𝑧(𝜓 → 𝜑))) |
9 | 3, 4, 8 | 3bitr4i 302 | 1 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ⊆ wss 3944 {coprab 7394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2370 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-oprab 7397 |
This theorem is referenced by: oprabbi 36832 |
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