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Theorem ss2rab 3223
Description: Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
Assertion
Ref Expression
ss2rab ({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))

Proof of Theorem ss2rab
StepHypRef Expression
1 df-rab 2457 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 df-rab 2457 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
31, 2sseq12i 3175 . 2 ({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥 ∣ (𝑥𝐴𝜓)})
4 ss2ab 3215 . 2 ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥 ∣ (𝑥𝐴𝜓)} ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
5 df-ral 2453 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
6 imdistan 442 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
76albii 1463 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
85, 7bitr2i 184 . 2 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) ↔ ∀𝑥𝐴 (𝜑𝜓))
93, 4, 83bitri 205 1 ({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1346  wcel 2141  {cab 2156  wral 2448  {crab 2452  wss 3121
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-in 3127  df-ss 3134
This theorem is referenced by:  ss2rabdv  3228  ss2rabi  3229
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