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Theorem rabbiia 2592
Description: Equivalent wff's yield equal restricted class abstractions (inference rule). (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 2195 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴𝜓)}
4 df-rab 2358 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
5 df-rab 2358 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
63, 4, 53eqtr4i 2112 1 {𝑥𝐴𝜑} = {𝑥𝐴𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  {cab 2068  {crab 2353
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-rab 2358
This theorem is referenced by:  bm2.5ii  4248  fndmdifcom  5305  cauappcvgprlemladdru  6908  cauappcvgprlemladdrl  6909  cauappcvgpr  6914  caucvgprlemcl  6928  caucvgprlemladdrl  6930  caucvgpr  6934  caucvgprprlemclphr  6957  ioopos  9049  gcdcom  10509  gcdass  10548  lcmcom  10590  lcmass  10611
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