Theorem reueq1f 2497

Description: Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)

Hypotheses

Ref	Expression
raleq1f.1	⊢ ℲxA
raleq1f.2	⊢ ℲxB

Assertion

Ref	Expression
reueq1f	⊢ (A = B → (∃!x ∈ A φ ↔ ∃!x ∈ B φ))

Proof of Theorem reueq1f

Step	Hyp	Ref	Expression
1		raleq1f.1	. . . 4 ⊢ ℲxA
2		raleq1f.2	. . . 4 ⊢ ℲxB
3	1, 2	nfeq 2182	. . 3 ⊢ Ⅎx A = B
4		eleq2 2098	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
5	4	anbi1d 438	. . 3 ⊢ (A = B → ((x ∈ A ∧ φ) ↔ (x ∈ B ∧ φ)))
6	3, 5	eubid 1904	. 2 ⊢ (A = B → (∃!x(x ∈ A ∧ φ) ↔ ∃!x(x ∈ B ∧ φ)))
7		df-reu 2307	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
8		df-reu 2307	. 2 ⊢ (∃!x ∈ B φ ↔ ∃!x(x ∈ B ∧ φ))
9	6, 7, 8	3bitr4g 212	1 ⊢ (A = B → (∃!x ∈ A φ ↔ ∃!x ∈ B φ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∃!weu 1897  Ⅎwnfc 2162  ∃!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-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-cleq 2030  df-clel 2033  df-nfc 2164  df-reu 2307

This theorem is referenced by:  reueq1  2501