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Theorem rabbidva 2542
 Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
Hypothesis
Ref Expression
rabbidva.1 ((φ x A) → (ψχ))
Assertion
Ref Expression
rabbidva (φ → {x Aψ} = {x Aχ})
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   χ(x)   A(x)

Proof of Theorem rabbidva
StepHypRef Expression
1 rabbidva.1 . . 3 ((φ x A) → (ψχ))
21ralrimiva 2386 . 2 (φx A (ψχ))
3 rabbi 2481 . 2 (x A (ψχ) ↔ {x Aψ} = {x Aχ})
42, 3sylib 127 1 (φ → {x Aψ} = {x Aχ})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  {crab 2304 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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-ral 2305  df-rab 2309 This theorem is referenced by:  rabbidv  2543  rabeqbidva  2547  rabbi2dva  3139  rabxfrd  4167  onsucmin  4198  seinxp  4354  fniniseg2  5232  fnniniseg2  5233  f1oresrab  5272
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