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Theorem rabeqf 2528
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 Aφ} = {x Bφ})

Proof of Theorem rabeqf
StepHypRef Expression
1 rabeqf.1 . . . 4 xA
2 rabeqf.2 . . . 4 xB
31, 2nfeq 2167 . . 3 x A = B
4 eleq2 2083 . . . 4 (A = B → (x Ax B))
54anbi1d 441 . . 3 (A = B → ((x A φ) ↔ (x B φ)))
63, 5abbid 2136 . 2 (A = B → {x ∣ (x A φ)} = {x ∣ (x B φ)})
7 df-rab 2293 . 2 {x Aφ} = {x ∣ (x A φ)}
8 df-rab 2293 . 2 {x Bφ} = {x ∣ (x B φ)}
96, 7, 83eqtr4g 2079 1 (A = B → {x Aφ} = {x Bφ})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228   wcel 1374  {cab 2008  wnfc 2147  {crab 2288
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rab 2293
This theorem is referenced by:  rabeq  2529
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