Theorem rmob2 3497

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 2904	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
4	2, 3	sylib 207	. . 3 ⊢ (𝜑 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
5		rmoi2.4	. . 3 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
6		rmoi2.5	. . 3 ⊢ (𝜑 → 𝜓)
7		eleq1 2676	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
8		rmoi2.1	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
9	7, 8	anbi12d 743	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝐵 ∈ 𝐴 ∧ 𝜒)))
10	9	mob2 3353	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → (𝑥 = 𝐵 ↔ (𝐵 ∈ 𝐴 ∧ 𝜒)))
11	1, 4, 5, 6, 10	syl112anc 1322	. 2 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ (𝐵 ∈ 𝐴 ∧ 𝜒)))
12	1, 11	mpbirand 529	1 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃*wmo 2459  ∃*wrmo 2899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rmo 2904  df-v 3175

This theorem is referenced by:  rmoi2  3498