Theorem rmoi2 3855

Description: Consequence of "restricted at most one". (Contributed by Thierry Arnoux, 9-Dec-2019.)

Hypotheses

Ref	Expression
rmoi2.1	⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
rmoi2.2	⊢ (𝜑 → 𝐵 ∈ 𝐴)
rmoi2.3	⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓)
rmoi2.4	⊢ (𝜑 → 𝑥 ∈ 𝐴)
rmoi2.5	⊢ (𝜑 → 𝜓)
rmoi2.6	⊢ (𝜑 → 𝜒)

Assertion

Ref	Expression
rmoi2	⊢ (𝜑 → 𝑥 = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rmoi2

Step	Hyp	Ref	Expression
1		rmoi2.6	. 2 ⊢ (𝜑 → 𝜒)
2		rmoi2.1	. . 3 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
3		rmoi2.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
4		rmoi2.3	. . 3 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓)
5		rmoi2.4	. . 3 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
6		rmoi2.5	. . 3 ⊢ (𝜑 → 𝜓)
7	2, 3, 4, 5, 6	rmob2 3854	. 2 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒))
8	1, 7	mpbird 260	1 ⊢ (𝜑 → 𝑥 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  ∃*wrmo 3375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-rmo 3376  df-v 3465

This theorem is referenced by:  lmieu  29047