Theorem reuss 3485

Description: Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)

Assertion

Ref	Expression
reuss	⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reuss

Step	Hyp	Ref	Expression
1		idd 21	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜑))
2	1	rgen 2583	. . 3 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜑)
3		reuss2 3484	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜑)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ∃!𝑥 ∈ 𝐴 𝜑)
4	2, 3	mpanl2 435	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ∃!𝑥 ∈ 𝐴 𝜑)
5	4	3impb 1223	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ∃!wreu 2510   ⊆ wss 3197

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-ral 2513  df-rex 2514  df-reu 2515  df-in 3203  df-ss 3210

This theorem is referenced by:  riotass  5977