Theorem reuan 39829

Description: Introduction of a conjunct into restricted uniqueness quantifier, analogous to euan 2518. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Hypothesis

Ref	Expression
rmoanim.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
reuan	⊢ (∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem reuan

Step	Hyp	Ref	Expression
1		rmoanim.1	. . . . . 6 ⊢ Ⅎ𝑥𝜑
2		simpl 472	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
3	2	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝜑))
4	1, 3	rexlimi 3006	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → 𝜑)
5	4	adantr 480	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → 𝜑)
6		simpr 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
7	6	reximi 2994	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)
8	7	adantr 480	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → ∃𝑥 ∈ 𝐴 𝜓)
9		nfre1 2988	. . . . . 6 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)
10	4	adantr 480	. . . . . . . . 9 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → 𝜑)
11	10	a1d 25	. . . . . . . 8 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜑))
12	11	ancrd 575	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜓 → (𝜑 ∧ 𝜓)))
13	6, 12	impbid2 215	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → ((𝜑 ∧ 𝜓) ↔ 𝜓))
14	9, 13	rmobida 3106	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃*𝑥 ∈ 𝐴 𝜓))
15	14	biimpa 500	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → ∃*𝑥 ∈ 𝐴 𝜓)
16	5, 8, 15	jca32 556	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → (𝜑 ∧ (∃𝑥 ∈ 𝐴 𝜓 ∧ ∃*𝑥 ∈ 𝐴 𝜓)))
17		reu5 3136	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)))
18		reu5 3136	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ (∃𝑥 ∈ 𝐴 𝜓 ∧ ∃*𝑥 ∈ 𝐴 𝜓))
19	18	anbi2i 726	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 ∧ (∃𝑥 ∈ 𝐴 𝜓 ∧ ∃*𝑥 ∈ 𝐴 𝜓)))
20	16, 17, 19	3imtr4i 280	. 2 ⊢ (∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓))
21		ibar 524	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
22	21	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ (𝜑 ∧ 𝜓)))
23	1, 22	reubida 3101	. . 3 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)))
24	23	biimpa 500	. 2 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
25	20, 24	impbii 198	1 ⊢ (∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  Ⅎwnf 1699   ∈ wcel 1977  ∃wrex 2897  ∃!wreu 2898  ∃*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-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904

This theorem is referenced by:  2reu7  39840  2reu8  39841