Theorem rmob2 3846

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 3369	. . . 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 2851	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
8		rmoi2.1	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
9	7, 8	anbi12d 643	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝐵 ∈ 𝐴 ∧ 𝜒)))
10	9	mob2 3678	. . 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 719	1 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃*wmo 2565  ∃*wrmo 3368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-rmo 3369  df-v 3457

This theorem is referenced by:  rmoi2  3847