Theorem rmob 3889

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 3379	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		simprl 770	. . . 4 ⊢ ((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → 𝐵 ∈ 𝐴)
3		eleq1 2828	. . . 4 ⊢ (𝐵 = 𝐶 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
4	2, 3	syl5ibcom 245	. . 3 ⊢ ((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 → 𝐶 ∈ 𝐴))
5		simpl 482	. . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝜒) → 𝐶 ∈ 𝐴)
6	5	a1i 11	. . 3 ⊢ ((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → ((𝐶 ∈ 𝐴 ∧ 𝜒) → 𝐶 ∈ 𝐴))
7	2	anim1i 615	. . . . 5 ⊢ (((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) ∧ 𝐶 ∈ 𝐴) → (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴))
8		simpll 766	. . . . 5 ⊢ (((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) ∧ 𝐶 ∈ 𝐴) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
9		simplr 768	. . . . 5 ⊢ (((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) ∧ 𝐶 ∈ 𝐴) → (𝐵 ∈ 𝐴 ∧ 𝜓))
10		eleq1 2828	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
11		rmoi.b	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
12	10, 11	anbi12d 632	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝐵 ∈ 𝐴 ∧ 𝜓)))
13		eleq1 2828	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
14		rmoi.c	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒))
15	13, 14	anbi12d 632	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))
16	12, 15	mob 3722	. . . . 5 ⊢ (((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))
17	7, 8, 9, 16	syl3anc 1372	. . . 4 ⊢ (((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))
18	17	ex 412	. . 3 ⊢ ((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐶 ∈ 𝐴 → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒))))
19	4, 6, 18	pm5.21ndd 379	. 2 ⊢ ((∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))
20	1, 19	sylanb 581	1 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∃*wmo 2537  ∃*wrmo 3378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-rmo 3379  df-v 3481

This theorem is referenced by:  rmoi  3890