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

Theorem bj-rabeqd 33222
 Description: Deduction form of rabeq 3332. Note that contrary to rabeq 3332 it has no dv condition. (Contributed by BJ, 27-Apr-2019.)
Hypotheses
Ref Expression
bj-rabeqd.nf 𝑥𝜑
bj-rabeqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
bj-rabeqd (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})

Proof of Theorem bj-rabeqd
StepHypRef Expression
1 bj-rabeqd.nf . 2 𝑥𝜑
2 bj-rabeqd.1 . . 3 (𝜑𝐴 = 𝐵)
3 eleq2 2828 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
43anbi1d 743 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜓)))
52, 4syl 17 . 2 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜓)))
61, 5bj-rabbida2 33219 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632  Ⅎwnf 1857   ∈ wcel 2139  {crab 3054 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-rab 3059 This theorem is referenced by:  bj-rabeqbid  33223  bj-rabeqbida  33224  bj-inrab2  33230
 Copyright terms: Public domain W3C validator