Theorem reupick3 3407

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 2451	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-rex 2450	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)))
3		anass 399	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)))
4	3	exbii 1593	. . . . 5 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)))
5	2, 4	bitr4i 186	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓))
6		eupick 2093	. . . 4 ⊢ ((∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
7	1, 5, 6	syl2anb 289	. . 3 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
8	7	expd 256	. 2 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
9	8	3impia 1190	1 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 968  ∃wex 1480  ∃!weu 2014   ∈ wcel 2136  ∃wrex 2445  ∃!wreu 2446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-3an 970  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-rex 2450  df-reu 2451

This theorem is referenced by:  reupick2  3408