Theorem reupick2 4306

Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Assertion

Ref	Expression
reupick2	⊢ (((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓))

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem reupick2

Step	Hyp	Ref	Expression
1		ancr 546	. . . . . 6 ⊢ ((𝜓 → 𝜑) → (𝜓 → (𝜑 ∧ 𝜓)))
2	1	ralimi 3073	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → ∀𝑥 ∈ 𝐴 (𝜓 → (𝜑 ∧ 𝜓)))
3		rexim 3077	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → (𝜑 ∧ 𝜓)) → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)))
4	2, 3	syl 17	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)))
5		reupick3 4305	. . . . . 6 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓))
6	5	3exp 1119	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))))
7	6	com12 32	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (∃!𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))))
8	4, 7	syl6 35	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → (∃𝑥 ∈ 𝐴 𝜓 → (∃!𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))))
9	8	3imp1 1348	. 2 ⊢ (((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓))
10		rsp 3230	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑)))
11	10	3ad2ant1 1133	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑)))
12	11	imp 406	. 2 ⊢ (((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜑))
13	9, 12	impbid 212	1 ⊢ (((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  ∃!wreu 3357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1780  df-mo 2539  df-eu 2568  df-ral 3052  df-rex 3061  df-reu 3360

This theorem is referenced by:  grpoidval  30494  grpoidinv2  30496  grpoinv  30506