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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 1434 | . . . 4 ⊢ Ⅎ𝑥∀𝑥∀𝑦(𝜑 → 𝜓) | |
2 | nfa1 1434 | . . . . . 6 ⊢ Ⅎ𝑦∀𝑦(𝜑 → 𝜓) | |
3 | sp 1401 | . . . . . . 7 ⊢ (∀𝑦(𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
4 | 3 | anim2d 320 | . . . . . 6 ⊢ (∀𝑦(𝜑 → 𝜓) → ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
5 | 2, 4 | eximd 1503 | . . . . 5 ⊢ (∀𝑦(𝜑 → 𝜓) → (∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
6 | 5 | sps 1430 | . . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
7 | 1, 6 | eximd 1503 | . . 3 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
8 | 7 | ss2abdv 3013 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} ⊆ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)}) |
9 | df-opab 3819 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
10 | df-opab 3819 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
11 | 8, 9, 10 | 3sstr4g 2986 | 1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 = wceq 1243 ∃wex 1381 {cab 2026 ⊆ wss 2917 〈cop 3378 {copab 3817 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-in 2924 df-ss 2931 df-opab 3819 |
This theorem is referenced by: ssopab2b 4013 ssopab2i 4014 ssopab2dv 4015 |
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