Theorem rmob2 3847

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)

Assertion

Ref	Expression
rmob2	⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒))

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

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

Proof of Theorem rmob2

Step	Hyp	Ref	Expression
1		rmoi2.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
2		rmoi2.3	. . . 4 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓)
3		df-rmo 3371	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
4	2, 3	sylib 221	. . 3 ⊢ (𝜑 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
5		rmoi2.4	. . 3 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
6		rmoi2.5	. . 3 ⊢ (𝜑 → 𝜓)
7		eleq1 2853	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
8		rmoi2.1	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
9	7, 8	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝐵 ∈ 𝐴 ∧ 𝜒)))
10	9	mob2 3680	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → (𝑥 = 𝐵 ↔ (𝐵 ∈ 𝐴 ∧ 𝜒)))
11	1, 4, 5, 6, 10	syl112anc 1401	. 2 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ (𝐵 ∈ 𝐴 ∧ 𝜒)))
12	1, 11	mpbirand 720	1 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃*wmo 2567  ∃*wrmo 3370

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-rmo 3371  df-v 3459

This theorem is used by:  rmoi2  3848