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Theorem rabbiia 2666
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 449 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32abbii 2253 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴𝜓)}
4 df-rab 2423 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
5 df-rab 2423 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
63, 4, 53eqtr4i 2168 1 {𝑥𝐴𝜑} = {𝑥𝐴𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  {cab 2123  {crab 2418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-rab 2423
This theorem is referenced by:  rabbii  2667  bm2.5ii  4407  fndmdifcom  5519  cauappcvgprlemladdru  7457  cauappcvgprlemladdrl  7458  cauappcvgpr  7463  caucvgprlemcl  7477  caucvgprlemladdrl  7479  caucvgpr  7483  caucvgprprlemclphr  7506  ioopos  9726  gcdcom  11651  gcdass  11692  lcmcom  11734  lcmass  11755
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