Theorem reu6i 2921

Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)

Assertion

Ref	Expression
reu6i	⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reu6i

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2180	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐵))
2	1	bibi2d 231	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ (𝜑 ↔ 𝑥 = 𝐵)))
3	2	ralbidv 2470	. . 3 ⊢ (𝑦 = 𝐵 → (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)))
4	3	rspcev 2834	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
5		reu6 2919	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
6	4, 5	sylibr 133	1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449  ∃!wreu 2450

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-v 2732

This theorem is referenced by:  eqreu  2922  riota5f  5833  negeu  8110  creur  8875  creui  8876