Theorem reupick2 3449

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 321	. . . . . 6 ⊢ ((𝜓 → 𝜑) → (𝜓 → (𝜑 ∧ 𝜓)))
2	1	ralimi 2560	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → ∀𝑥 ∈ 𝐴 (𝜓 → (𝜑 ∧ 𝜓)))
3		rexim 2591	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → (𝜑 ∧ 𝜓)) → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)))
4	2, 3	syl 14	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)))
5		reupick3 3448	. . . . . 6 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓))
6	5	3exp 1204	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))))
7	6	com12 30	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (∃!𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))))
8	4, 7	syl6 33	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → (∃𝑥 ∈ 𝐴 𝜓 → (∃!𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))))
9	8	3imp1 1222	. 2 ⊢ (((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓))
10		rsp 2544	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑)))
11	10	3ad2ant1 1020	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑)))
12	11	imp 124	. 2 ⊢ (((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜑))
13	9, 12	impbid 129	1 ⊢ (((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  ∃!wreu 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-ral 2480  df-rex 2481  df-reu 2482

This theorem is referenced by: (None)