Theorem eqreu 2727

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 2441	. . . . 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 2579	. . . . . 6 ⊢ (B ∈ A → (∀x ∈ A (x = B → φ) ↔ ψ))
4	3	anbi2d 437	. . . . 5 ⊢ (B ∈ A → ((∀x ∈ A (φ → x = B) ∧ ∀x ∈ A (x = B → φ)) ↔ (∀x ∈ A (φ → x = B) ∧ ψ)))
5	1, 4	syl5bb 181	. . . 4 ⊢ (B ∈ A → (∀x ∈ A (φ ↔ x = B) ↔ (∀x ∈ A (φ → x = B) ∧ ψ)))
6		reu6i 2726	. . . . 5 ⊢ ((B ∈ A ∧ ∀x ∈ A (φ ↔ x = B)) → ∃!x ∈ A φ)
7	6	ex 108	. . . 4 ⊢ (B ∈ A → (∀x ∈ A (φ ↔ x = B) → ∃!x ∈ A φ))
8	5, 7	sylbird 159	. . 3 ⊢ (B ∈ A → ((∀x ∈ A (φ → x = B) ∧ ψ) → ∃!x ∈ A φ))
9	8	3impib 1101	. 2 ⊢ ((B ∈ A ∧ ∀x ∈ A (φ → x = B) ∧ ψ) → ∃!x ∈ A φ)
10	9	3com23 1109	1 ⊢ ((B ∈ A ∧ ψ ∧ ∀x ∈ A (φ → x = B)) → ∃!x ∈ A φ)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∃!wreu 2302

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553

This theorem is referenced by: (None)