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Mirrors > Home > ILE Home > Th. List > ssopab2 | GIF version |
Description: Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.) |
Ref | Expression |
---|---|
ssopab2 | ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 1506 | . . . 4 ⊢ Ⅎ𝑥∀𝑥∀𝑦(𝜑 → 𝜓) | |
2 | nfa1 1506 | . . . . . 6 ⊢ Ⅎ𝑦∀𝑦(𝜑 → 𝜓) | |
3 | sp 1473 | . . . . . . 7 ⊢ (∀𝑦(𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
4 | 3 | anim2d 335 | . . . . . 6 ⊢ (∀𝑦(𝜑 → 𝜓) → ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
5 | 2, 4 | eximd 1576 | . . . . 5 ⊢ (∀𝑦(𝜑 → 𝜓) → (∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
6 | 5 | sps 1502 | . . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
7 | 1, 6 | eximd 1576 | . . 3 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
8 | 7 | ss2abdv 3140 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} ⊆ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)}) |
9 | df-opab 3960 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
10 | df-opab 3960 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
11 | 8, 9, 10 | 3sstr4g 3110 | 1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1314 = wceq 1316 ∃wex 1453 {cab 2103 ⊆ wss 3041 〈cop 3500 {copab 3958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-in 3047 df-ss 3054 df-opab 3960 |
This theorem is referenced by: ssopab2b 4168 ssopab2i 4169 ssopab2dv 4170 |
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