Theorem rmob2 3824

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 3344	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
4	2, 3	sylib 219	. . 3 ⊢ (𝜑 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
5		rmoi2.4	. . 3 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
6		rmoi2.5	. . 3 ⊢ (𝜑 → 𝜓)
7		eleq1 2827	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
8		rmoi2.1	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
9	7, 8	anbi12d 638	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝐵 ∈ 𝐴 ∧ 𝜒)))
10	9	mob2 3656	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → (𝑥 = 𝐵 ↔ (𝐵 ∈ 𝐴 ∧ 𝜒)))
11	1, 4, 5, 6, 10	syl112anc 1382	. 2 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ (𝐵 ∈ 𝐴 ∧ 𝜒)))
12	1, 11	mpbirand 713	1 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃*wmo 2541  ∃*wrmo 3343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-clab 2718  df-cleq 2731  df-clel 2814  df-rmo 3344  df-v 3433

This theorem is referenced by:  rmoi2  3825