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Theorem rmoeq1f 3167
Description: Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypotheses
Ref Expression
raleq1f.1 𝑥𝐴
raleq1f.2 𝑥𝐵
Assertion
Ref Expression
rmoeq1f (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))

Proof of Theorem rmoeq1f
StepHypRef Expression
1 raleq1f.1 . . . 4 𝑥𝐴
2 raleq1f.2 . . . 4 𝑥𝐵
31, 2nfeq 2805 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2719 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54anbi1d 741 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5mobid 2517 . 2 (𝐴 = 𝐵 → (∃*𝑥(𝑥𝐴𝜑) ↔ ∃*𝑥(𝑥𝐵𝜑)))
7 df-rmo 2949 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
8 df-rmo 2949 . 2 (∃*𝑥𝐵 𝜑 ↔ ∃*𝑥(𝑥𝐵𝜑))
96, 7, 83bitr4g 303 1 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  ∃*wmo 2499  wnfc 2780  ∃*wrmo 2944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-eu 2502  df-mo 2503  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rmo 2949
This theorem is referenced by:  rmoeq1  3171
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