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| Mirrors > Home > MPE Home > Th. List > ssoprab2b | Structured version Visualization version GIF version | ||
| Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 5554. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| ssoprab2b | ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfoprab1 7494 | . . . 4 ⊢ Ⅎ𝑥{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
| 2 | nfoprab1 7494 | . . . 4 ⊢ Ⅎ𝑥{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} | |
| 3 | 1, 2 | nfss 3976 | . . 3 ⊢ Ⅎ𝑥{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
| 4 | nfoprab2 7495 | . . . . 5 ⊢ Ⅎ𝑦{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
| 5 | nfoprab2 7495 | . . . . 5 ⊢ Ⅎ𝑦{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} | |
| 6 | 4, 5 | nfss 3976 | . . . 4 ⊢ Ⅎ𝑦{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
| 7 | nfoprab3 7496 | . . . . . 6 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
| 8 | nfoprab3 7496 | . . . . . 6 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} | |
| 9 | 7, 8 | nfss 3976 | . . . . 5 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
| 10 | ssel 3977 | . . . . . 6 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓})) | |
| 11 | oprabid 7463 | . . . . . 6 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜑) | |
| 12 | oprabid 7463 | . . . . . 6 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ 𝜓) | |
| 13 | 10, 11, 12 | 3imtr3g 295 | . . . . 5 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → (𝜑 → 𝜓)) |
| 14 | 9, 13 | alrimi 2213 | . . . 4 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → ∀𝑧(𝜑 → 𝜓)) |
| 15 | 6, 14 | alrimi 2213 | . . 3 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → ∀𝑦∀𝑧(𝜑 → 𝜓)) |
| 16 | 3, 15 | alrimi 2213 | . 2 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → ∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓)) |
| 17 | ssoprab2 7501 | . 2 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓}) | |
| 18 | 16, 17 | impbii 209 | 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 206 ∀wal 1538 ∈ wcel 2108 ⊆ wss 3951 〈cop 4632 {coprab 7432 |
| 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-oprab 7435 |
| This theorem is referenced by: eqoprab2b 7504 |
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