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Theorem rabbiia 2604
Description: Equivalent wff's yield equal restricted class abstractions (inference form). (Contributed by NM, 22-May-1999.)
Hypothesis
Ref Expression
rabbiia.1 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rabbiia {𝑥𝐴𝜑} = {𝑥𝐴𝜓}

Proof of Theorem rabbiia
StepHypRef Expression
1 rabbiia.1 . . . 4 (𝑥𝐴 → (𝜑𝜓))
21pm5.32i 442 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32abbii 2203 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴𝜓)}
4 df-rab 2368 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
5 df-rab 2368 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
63, 4, 53eqtr4i 2118 1 {𝑥𝐴𝜑} = {𝑥𝐴𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  {cab 2074  {crab 2363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-rab 2368
This theorem is referenced by:  rabbii  2605  bm2.5ii  4303  fndmdifcom  5389  cauappcvgprlemladdru  7194  cauappcvgprlemladdrl  7195  cauappcvgpr  7200  caucvgprlemcl  7214  caucvgprlemladdrl  7216  caucvgpr  7220  caucvgprprlemclphr  7243  ioopos  9337  gcdcom  11058  gcdass  11097  lcmcom  11139  lcmass  11160
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