Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-rabbida2 Structured version   Visualization version   GIF version

Theorem bj-rabbida2 32105
 Description: Version of rabbidva2 3162 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
Hypotheses
Ref Expression
bj-rabbida2.nf 𝑥𝜑
bj-rabbida2.1 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
Assertion
Ref Expression
bj-rabbida2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Proof of Theorem bj-rabbida2
StepHypRef Expression
1 bj-rabbida2.nf . . 3 𝑥𝜑
2 bj-rabbida2.1 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
31, 2abbid 2727 . 2 (𝜑 → {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐵𝜒)})
4 df-rab 2905 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
5 df-rab 2905 . 2 {𝑥𝐵𝜒} = {𝑥 ∣ (𝑥𝐵𝜒)}
63, 4, 53eqtr4g 2669 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  {cab 2596  {crab 2900 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-rab 2905 This theorem is referenced by:  bj-rabeqd  32108
 Copyright terms: Public domain W3C validator