Theorem reupick3 3492

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

Assertion

Ref	Expression
reupick3	⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓))

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem reupick3

Step	Hyp	Ref	Expression
1		df-reu 2517	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-rex 2516	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)))
3		anass 401	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)))
4	3	exbii 1653	. . . . 5 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)))
5	2, 4	bitr4i 187	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓))
6		eupick 2159	. . . 4 ⊢ ((∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
7	1, 5, 6	syl2anb 291	. . 3 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
8	7	expd 258	. 2 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
9	8	3impia 1226	1 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004  ∃wex 1540  ∃!weu 2079   ∈ wcel 2202  ∃wrex 2511  ∃!wreu 2512

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-rex 2516  df-reu 2517

This theorem is referenced by:  reupick2  3493