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Mirrors > Home > MPE Home > Th. List > ssopab2bw | Structured version Visualization version GIF version |
Description: Equivalence of ordered pair abstraction subclass and implication. Version of ssopab2b 5429 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 27-Dec-1996.) (Revised by Gino Giotto, 26-Jan-2024.) |
Ref | Expression |
---|---|
ssopab2bw | ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfopab1 5128 | . . . 4 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | nfopab1 5128 | . . . 4 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜓} | |
3 | 1, 2 | nfss 3953 | . . 3 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} |
4 | nfopab2 5129 | . . . . 5 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
5 | nfopab2 5129 | . . . . 5 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜓} | |
6 | 4, 5 | nfss 3953 | . . . 4 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} |
7 | ssel 3954 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} → (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓})) | |
8 | opabidw 5405 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
9 | opabidw 5405 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ 𝜓) | |
10 | 7, 8, 9 | 3imtr3g 297 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} → (𝜑 → 𝜓)) |
11 | 6, 10 | alrimi 2212 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} → ∀𝑦(𝜑 → 𝜓)) |
12 | 3, 11 | alrimi 2212 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} → ∀𝑥∀𝑦(𝜑 → 𝜓)) |
13 | ssopab2 5426 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓}) | |
14 | 12, 13 | impbii 211 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1534 ∈ wcel 2113 ⊆ wss 3929 〈cop 4566 {copab 5121 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5122 |
This theorem is referenced by: eqopab2bw 5428 dffun2 6358 marypha2lem3 8894 cvmlift2lem12 32582 cossssid2 35741 |
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