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Theorem rabbiia 2541
 Description: Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
Hypothesis
Ref Expression
rabbiia.1 (x A → (φψ))
Assertion
Ref Expression
rabbiia {x Aφ} = {x Aψ}

Proof of Theorem rabbiia
StepHypRef Expression
1 rabbiia.1 . . . 4 (x A → (φψ))
21pm5.32i 427 . . 3 ((x A φ) ↔ (x A ψ))
32abbii 2150 . 2 {x ∣ (x A φ)} = {x ∣ (x A ψ)}
4 df-rab 2309 . 2 {x Aφ} = {x ∣ (x A φ)}
5 df-rab 2309 . 2 {x Aψ} = {x ∣ (x A ψ)}
63, 4, 53eqtr4i 2067 1 {x Aφ} = {x Aψ}
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {cab 2023  {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-rab 2309 This theorem is referenced by:  bm2.5ii  4188  fndmdifcom  5216  cauappcvgprlemladdru  6628  cauappcvgprlemladdrl  6629  cauappcvgpr  6634  caucvgprlemcl  6647  caucvgprlemladdrl  6649  caucvgpr  6653  ioopos  8589
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