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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 5429. Usage of this theorem is discouraged because it depends on ax-13 2386. (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 7209 | . . . 4 ⊢ Ⅎ𝑥{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
2 | nfoprab1 7209 | . . . 4 ⊢ Ⅎ𝑥{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} | |
3 | 1, 2 | nfss 3960 | . . 3 ⊢ Ⅎ𝑥{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
4 | nfoprab2 7210 | . . . . 5 ⊢ Ⅎ𝑦{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
5 | nfoprab2 7210 | . . . . 5 ⊢ Ⅎ𝑦{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} | |
6 | 4, 5 | nfss 3960 | . . . 4 ⊢ Ⅎ𝑦{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
7 | nfoprab3 7211 | . . . . . 6 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
8 | nfoprab3 7211 | . . . . . 6 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} | |
9 | 7, 8 | nfss 3960 | . . . . 5 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
10 | ssel 3961 | . . . . . 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 7182 | . . . . . 6 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜑) | |
12 | oprabid 7182 | . . . . . 6 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ 𝜓) | |
13 | 10, 11, 12 | 3imtr3g 297 | . . . . 5 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → (𝜑 → 𝜓)) |
14 | 9, 13 | alrimi 2208 | . . . 4 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → ∀𝑧(𝜑 → 𝜓)) |
15 | 6, 14 | alrimi 2208 | . . 3 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → ∀𝑦∀𝑧(𝜑 → 𝜓)) |
16 | 3, 15 | alrimi 2208 | . 2 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → ∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓)) |
17 | ssoprab2 7216 | . 2 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓}) | |
18 | 16, 17 | impbii 211 | 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 208 ∀wal 1531 ∈ wcel 2110 ⊆ wss 3936 〈cop 4567 {coprab 7151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-13 2386 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-oprab 7154 |
This theorem is referenced by: eqoprab2b 7219 |
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