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| Description: Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.) | 
| Ref | Expression | 
|---|---|
| reuss | ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . . . 4 ⊢ (𝜑 → 𝜑) | |
| 2 | 1 | rgenw 3064 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜑) | 
| 3 | reuss2 4325 | . . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜑)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ∃!𝑥 ∈ 𝐴 𝜑) | |
| 4 | 2, 3 | mpanl2 701 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ∃!𝑥 ∈ 𝐴 𝜑) | 
| 5 | 4 | 3impb 1114 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∀wral 3060 ∃wrex 3069 ∃!wreu 3377 ⊆ wss 3950 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-ex 1779 df-mo 2539 df-eu 2568 df-clel 2815 df-ral 3061 df-rex 3070 df-reu 3380 df-ss 3967 | 
| This theorem is referenced by: euelss 4331 riotass 7420 adjbdln 32103 tfsconcatlem 43354 tfsconcatfv 43359 | 
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