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Theorem rabbi 2465
Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2526. (Contributed by NM, 25-Nov-2013.)
Assertion
Ref Expression
rabbi (x A (ψχ) ↔ {x Aψ} = {x Aχ})

Proof of Theorem rabbi
StepHypRef Expression
1 abbi 2133 . 2 (x((x A ψ) ↔ (x A χ)) ↔ {x ∣ (x A ψ)} = {x ∣ (x A χ)})
2 df-ral 2289 . . 3 (x A (ψχ) ↔ x(x A → (ψχ)))
3 pm5.32 429 . . . 4 ((x A → (ψχ)) ↔ ((x A ψ) ↔ (x A χ)))
43albii 1339 . . 3 (x(x A → (ψχ)) ↔ x((x A ψ) ↔ (x A χ)))
52, 4bitri 173 . 2 (x A (ψχ) ↔ x((x A ψ) ↔ (x A χ)))
6 df-rab 2293 . . 3 {x Aψ} = {x ∣ (x A ψ)}
7 df-rab 2293 . . 3 {x Aχ} = {x ∣ (x A χ)}
86, 7eqeq12i 2035 . 2 ({x Aψ} = {x Aχ} ↔ {x ∣ (x A ψ)} = {x ∣ (x A χ)})
91, 5, 83bitr4i 201 1 (x A (ψχ) ↔ {x Aψ} = {x Aχ})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1226   = wceq 1228   wcel 1374  {cab 2008  wral 2284  {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-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  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-ral 2289  df-rab 2293
This theorem is referenced by:  rabbidva  2526
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