Theorem reuan 3886

Description: Introduction of a conjunct into restricted unique existential quantifier, analogous to euan 2616. (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))

Proof of Theorem reuan

Step	Hyp	Ref	Expression
1		rmoanim.1	. . . . . 6 ⊢ Ⅎ𝑥𝜑
2		simpl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
3	2	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝜑))
4	1, 3	rexlimi 3255	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → 𝜑)
5	4	adantr 481	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → 𝜑)
6		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
7	6	reximi 3083	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)
8	7	adantr 481	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → ∃𝑥 ∈ 𝐴 𝜓)
9		nfre1 3281	. . . . . 6 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)
10	4	adantr 481	. . . . . . . . 9 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → 𝜑)
11	10	a1d 25	. . . . . . . 8 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜑))
12	11	ancrd 552	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜓 → (𝜑 ∧ 𝜓)))
13	6, 12	impbid2 225	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → ((𝜑 ∧ 𝜓) ↔ 𝜓))
14	9, 13	rmobida 3401	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃*𝑥 ∈ 𝐴 𝜓))
15	14	biimpa 477	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → ∃*𝑥 ∈ 𝐴 𝜓)
16	5, 8, 15	jca32 516	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → (𝜑 ∧ (∃𝑥 ∈ 𝐴 𝜓 ∧ ∃*𝑥 ∈ 𝐴 𝜓)))
17		reu5 3377	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)))
18		reu5 3377	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ (∃𝑥 ∈ 𝐴 𝜓 ∧ ∃*𝑥 ∈ 𝐴 𝜓))
19	18	anbi2i 623	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 ∧ (∃𝑥 ∈ 𝐴 𝜓 ∧ ∃*𝑥 ∈ 𝐴 𝜓)))
20	16, 17, 19	3imtr4i 291	. 2 ⊢ (∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓))
21		ibar 529	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
22	21	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ (𝜑 ∧ 𝜓)))
23	1, 22	reubida 3402	. . 3 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)))
24	23	biimpa 477	. 2 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
25	20, 24	impbii 208	1 ⊢ (∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  Ⅎwnf 1785   ∈ wcel 2106  ∃wrex 3069  ∃!wreu 3373  ∃*wrmo 3374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-nf 1786  df-mo 2533  df-eu 2562  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376

This theorem is referenced by:  2reu7  45589  2reu8  45590