Theorem reuss 4346

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		id 22	. . . 4 ⊢ (𝜑 → 𝜑)
2	1	rgenw 3071	. . 3 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜑)
3		reuss2 4345	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜑)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ∃!𝑥 ∈ 𝐴 𝜑)
4	2, 3	mpanl2 700	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ∃!𝑥 ∈ 𝐴 𝜑)
5	4	3impb 1115	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wral 3067  ∃wrex 3076  ∃!wreu 3386   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-ex 1778  df-mo 2543  df-eu 2572  df-clel 2819  df-ral 3068  df-rex 3077  df-reu 3389  df-ss 3993

This theorem is referenced by:  euelss  4351  riotass  7436  adjbdln  32115  tfsconcatlem  43298  tfsconcatfv  43303