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Theorem rabbidva2 2644
 Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
Hypothesis
Ref Expression
rabbidva2.1 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
Assertion
Ref Expression
rabbidva2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rabbidva2
StepHypRef Expression
1 rabbidva2.1 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
21abbidv 2232 . 2 (𝜑 → {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐵𝜒)})
3 df-rab 2399 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
4 df-rab 2399 . 2 {𝑥𝐵𝜒} = {𝑥 ∣ (𝑥𝐵𝜒)}
52, 3, 43eqtr4g 2172 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1314   ∈ wcel 1463  {cab 2101  {crab 2394 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-rab 2399 This theorem is referenced by: (None)
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