Theorem reu6i 3718

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 2833	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐵))
2	1	bibi2d 345	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ (𝜑 ↔ 𝑥 = 𝐵)))
3	2	ralbidv 3197	. . 3 ⊢ (𝑦 = 𝐵 → (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)))
4	3	rspcev 3622	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
5		reu6 3716	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
6	4, 5	sylibr 236	1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-cleq 2814  df-clel 2893  df-ral 3143  df-rex 3144  df-reu 3145

This theorem is referenced by:  eqreu  3719  riota5f  7141  negeu  10875  creur  11631  creui  11632  reuccatpfxs1  14108  lublecl  17598  dfod2  18690  lmieu  26569  esum2dlem  31351  fvineqsneu  34691  poimirlem16  34907  poimirlem17  34908  poimirlem19  34910  poimirlem20  34911  poimirlem22  34913  renegeulemv  39196