Theorem eqreu 3029

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

Hypothesis

Ref	Expression
eqreu.1	⊢ (x = B → (φ ↔ ψ))

Assertion

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

Distinct variable groups:   x,A   x,B   ψ,x

Allowed substitution hint:   φ(x)

Proof of Theorem eqreu

Step	Hyp	Ref	Expression
1		ralbiim 2752	. . . . 5 ⊢ (∀x ∈ A (φ ↔ x = B) ↔ (∀x ∈ A (φ → x = B) ∧ ∀x ∈ A (x = B → φ)))
2		eqreu.1	. . . . . . 7 ⊢ (x = B → (φ ↔ ψ))
3	2	ceqsralv 2887	. . . . . 6 ⊢ (B ∈ A → (∀x ∈ A (x = B → φ) ↔ ψ))
4	3	anbi2d 684	. . . . 5 ⊢ (B ∈ A → ((∀x ∈ A (φ → x = B) ∧ ∀x ∈ A (x = B → φ)) ↔ (∀x ∈ A (φ → x = B) ∧ ψ)))
5	1, 4	syl5bb 248	. . . 4 ⊢ (B ∈ A → (∀x ∈ A (φ ↔ x = B) ↔ (∀x ∈ A (φ → x = B) ∧ ψ)))
6		reu6i 3028	. . . . 5 ⊢ ((B ∈ A ∧ ∀x ∈ A (φ ↔ x = B)) → ∃!x ∈ A φ)
7	6	ex 423	. . . 4 ⊢ (B ∈ A → (∀x ∈ A (φ ↔ x = B) → ∃!x ∈ A φ))
8	5, 7	sylbird 226	. . 3 ⊢ (B ∈ A → ((∀x ∈ A (φ → x = B) ∧ ψ) → ∃!x ∈ A φ))
9	8	3impib 1149	. 2 ⊢ ((B ∈ A ∧ ∀x ∈ A (φ → x = B) ∧ ψ) → ∃!x ∈ A φ)
10	9	3com23 1157	1 ⊢ ((B ∈ A ∧ ψ ∧ ∀x ∈ A (φ → x = B)) → ∃!x ∈ A φ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃!wreu 2617

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-3an 936  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 2479  df-ral 2620  df-rex 2621  df-reu 2622  df-v 2862

This theorem is referenced by: (None)