MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rmoeq1 Structured version   Visualization version   GIF version

Theorem rmoeq1 3400
Description: Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2146, ax-11 2162, and ax-12 2179. (Revised by Steven Nguyen, 30-Apr-2023.)
Assertion
Ref Expression
rmoeq1 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rmoeq1
StepHypRef Expression
1 eleq2 2904 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21anbi1d 632 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
32mobidv 2634 . 2 (𝐴 = 𝐵 → (∃*𝑥(𝑥𝐴𝜑) ↔ ∃*𝑥(𝑥𝐵𝜑)))
4 df-rmo 3141 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
5 df-rmo 3141 . 2 (∃*𝑥𝐵 𝜑 ↔ ∃*𝑥(𝑥𝐵𝜑))
63, 4, 53bitr4g 317 1 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  ∃*wmo 2622  ∃*wrmo 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2624  df-cleq 2817  df-clel 2896  df-rmo 3141
This theorem is referenced by:  rmoeqd  3406  rmosn  4640  poimirlem2  34971
  Copyright terms: Public domain W3C validator