Theorem rabeq 2545

Description: Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)

Assertion

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

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   φ(x)

Proof of Theorem rabeq

Step	Hyp	Ref	Expression
1		nfcv 2175	. 2 ⊢ ℲxA
2		nfcv 2175	. 2 ⊢ ℲxB
3	1, 2	rabeqf 2544	1 ⊢ (A = B → {x ∈ A ∣ φ} = {x ∈ B ∣ φ})

Syntax hints:   → wi 4   = wceq 1242  {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:  rabeqbidv  2546  rabeqbidva  2547  difeq1  3049  ifeq1  3328  ifeq2  3329  iooval2  8554  fzval2  8647