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Theorem rabeqf 2852
 Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
Hypotheses
Ref Expression
rabeqf.1 xA
rabeqf.2 xB
Assertion
Ref Expression
rabeqf (A = B → {x A φ} = {x B φ})

Proof of Theorem rabeqf
StepHypRef Expression
1 rabeqf.1 . . . 4 xA
2 rabeqf.2 . . . 4 xB
31, 2nfeq 2496 . . 3 x A = B
4 eleq2 2414 . . . 4 (A = B → (x Ax B))
54anbi1d 685 . . 3 (A = B → ((x A φ) ↔ (x B φ)))
63, 5abbid 2466 . 2 (A = B → {x (x A φ)} = {x (x B φ)})
7 df-rab 2623 . 2 {x A φ} = {x (x A φ)}
8 df-rab 2623 . 2 {x B φ} = {x (x B φ)}
96, 7, 83eqtr4g 2410 1 (A = B → {x A φ} = {x B φ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476  {crab 2618 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623 This theorem is referenced by:  rabeq  2853
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