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Theorem ss2rab 3342
 Description: Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
Assertion
Ref Expression
ss2rab ({x A φ} {x A ψ} ↔ x A (φψ))

Proof of Theorem ss2rab
StepHypRef Expression
1 df-rab 2623 . . 3 {x A φ} = {x (x A φ)}
2 df-rab 2623 . . 3 {x A ψ} = {x (x A ψ)}
31, 2sseq12i 3297 . 2 ({x A φ} {x A ψ} ↔ {x (x A φ)} {x (x A ψ)})
4 ss2ab 3334 . 2 ({x (x A φ)} {x (x A ψ)} ↔ x((x A φ) → (x A ψ)))
5 df-ral 2619 . . 3 (x A (φψ) ↔ x(x A → (φψ)))
6 imdistan 670 . . . 4 ((x A → (φψ)) ↔ ((x A φ) → (x A ψ)))
76albii 1566 . . 3 (x(x A → (φψ)) ↔ x((x A φ) → (x A ψ)))
85, 7bitr2i 241 . 2 (x((x A φ) → (x A ψ)) ↔ x A (φψ))
93, 4, 83bitri 262 1 ({x A φ} {x A ψ} ↔ x A (φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  ∀wral 2614  {crab 2618   ⊆ wss 3257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  ss2rabdv  3347  ss2rabi  3348
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