Theorem reu6i 3696

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 2741	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐵))
2	1	bibi2d 342	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ (𝜑 ↔ 𝑥 = 𝐵)))
3	2	ralbidv 3156	. . 3 ⊢ (𝑦 = 𝐵 → (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)))
4	3	rspcev 3585	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
5		reu6 3694	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
6	4, 5	sylibr 234	1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  ∃!wreu 3349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-reu 3352

This theorem is referenced by:  eqreu  3697  riota5f  7354  negeu  11387  creur  12156  creui  12157  reuccatpfxs1  14688  lublecl  18300  dfod2  19478  lmieu  28764  reu6dv  32452  esum2dlem  34075  fvineqsneu  37392  poimirlem16  37623  poimirlem17  37624  poimirlem19  37626  poimirlem20  37627  poimirlem22  37629  renegeulemv  42349  sn-subeu  42408  upeu2lem  49010