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Theorem ss2rab 3010
 Description: Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
Assertion
Ref Expression
ss2rab ({x Aφ} ⊆ {x Aψ} ↔ x A (φψ))

Proof of Theorem ss2rab
StepHypRef Expression
1 df-rab 2309 . . 3 {x Aφ} = {x ∣ (x A φ)}
2 df-rab 2309 . . 3 {x Aψ} = {x ∣ (x A ψ)}
31, 2sseq12i 2965 . 2 ({x Aφ} ⊆ {x Aψ} ↔ {x ∣ (x A φ)} ⊆ {x ∣ (x A ψ)})
4 ss2ab 3002 . 2 ({x ∣ (x A φ)} ⊆ {x ∣ (x A ψ)} ↔ x((x A φ) → (x A ψ)))
5 df-ral 2305 . . 3 (x A (φψ) ↔ x(x A → (φψ)))
6 imdistan 418 . . . 4 ((x A → (φψ)) ↔ ((x A φ) → (x A ψ)))
76albii 1356 . . 3 (x(x A → (φψ)) ↔ x((x A φ) → (x A ψ)))
85, 7bitr2i 174 . 2 (x((x A φ) → (x A ψ)) ↔ x A (φψ))
93, 4, 83bitri 195 1 ({x Aφ} ⊆ {x Aψ} ↔ x A (φψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   ∈ wcel 1390  {cab 2023  ∀wral 2300  {crab 2304   ⊆ wss 2911 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-in 2918  df-ss 2925 This theorem is referenced by:  ss2rabdv  3015  ss2rabi  3016
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