Theorem rmob 3038

Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
rmoi.b	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
rmoi.c	⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
rmob	⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rmob

Step	Hyp	Ref	Expression
1		df-rmo 2450	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		simprl 521	. . . 4 ⊢ ((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → 𝐵 ∈ 𝐴)
3		eleq1 2227	. . . 4 ⊢ (𝐵 = 𝐶 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
4	2, 3	syl5ibcom 154	. . 3 ⊢ ((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 → 𝐶 ∈ 𝐴))
5		simpl 108	. . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝜒) → 𝐶 ∈ 𝐴)
6	5	a1i 9	. . 3 ⊢ ((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → ((𝐶 ∈ 𝐴 ∧ 𝜒) → 𝐶 ∈ 𝐴))
7		simplrl 525	. . . . 5 ⊢ (((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)
8		simpr 109	. . . . 5 ⊢ (((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐴)
9		simpll 519	. . . . 5 ⊢ (((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) ∧ 𝐶 ∈ 𝐴) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
10		simplrr 526	. . . . 5 ⊢ (((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) ∧ 𝐶 ∈ 𝐴) → 𝜓)
11		eleq1 2227	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
12		rmoi.b	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
13	11, 12	anbi12d 465	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝐵 ∈ 𝐴 ∧ 𝜓)))
14		eleq1 2227	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
15		rmoi.c	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒))
16	14, 15	anbi12d 465	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))
17	13, 16	mob 2903	. . . . 5 ⊢ (((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))
18	7, 8, 9, 7, 10, 17	syl212anc 1237	. . . 4 ⊢ (((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))
19	18	ex 114	. . 3 ⊢ ((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐶 ∈ 𝐴 → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒))))
20	4, 6, 19	pm5.21ndd 695	. 2 ⊢ ((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))
21	1, 20	sylanb 282	1 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1342  ∃*wmo 2014   ∈ wcel 2135  ∃*wrmo 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rmo 2450  df-v 2723

This theorem is referenced by:  rmoi  3039