Theorem rabeqf 2544

Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)

Hypotheses

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

Assertion

Ref	Expression
rabeqf	⊢ (A = B → {x ∈ A ∣ φ} = {x ∈ B ∣ φ})

Proof of Theorem rabeqf

Step	Hyp	Ref	Expression
1		rabeqf.1	. . . 4 ⊢ ℲxA
2		rabeqf.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	abbid 2151	. 2 ⊢ (A = B → {x ∣ (x ∈ A ∧ φ)} = {x ∣ (x ∈ B ∧ φ)})
7		df-rab 2309	. 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
8		df-rab 2309	. 2 ⊢ {x ∈ B ∣ φ} = {x ∣ (x ∈ B ∧ φ)}
9	6, 7, 8	3eqtr4g 2094	1 ⊢ (A = B → {x ∈ A ∣ φ} = {x ∈ B ∣ φ})

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  {cab 2023  Ⅎwnfc 2162  {crab 2304

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-bndl 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-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309

This theorem is referenced by:  rabeq  2545