Theorem rmoi2 3837

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 3836	. 2 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒))
8	1, 7	mpbird 256	1 ⊢ (𝜑 → 𝑥 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1540   ∈ wcel 2105  ∃*wrmo 3348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-rmo 3349  df-v 3443

This theorem is referenced by:  lmieu  27434