Theorem reu6i 3027

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 2362	. . . . 5 ⊢ (y = B → (x = y ↔ x = B))
2	1	bibi2d 309	. . . 4 ⊢ (y = B → ((φ ↔ x = y) ↔ (φ ↔ x = B)))
3	2	ralbidv 2634	. . 3 ⊢ (y = B → (∀x ∈ A (φ ↔ x = y) ↔ ∀x ∈ A (φ ↔ x = B)))
4	3	rspcev 2955	. 2 ⊢ ((B ∈ A ∧ ∀x ∈ A (φ ↔ x = B)) → ∃y ∈ A ∀x ∈ A (φ ↔ x = y))
5		reu6 3025	. 2 ⊢ (∃!x ∈ A φ ↔ ∃y ∈ A ∀x ∈ A (φ ↔ x = y))
6	4, 5	sylibr 203	1 ⊢ ((B ∈ A ∧ ∀x ∈ A (φ ↔ x = B)) → ∃!x ∈ A φ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-reu 2621  df-v 2861

This theorem is referenced by:  eqreu  3028