Theorem rabeq 2853

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 2489	. 2 ⊢ ℲxA
2		nfcv 2489	. 2 ⊢ ℲxB
3	1, 2	rabeqf 2852	1 ⊢ (A = B → {x ∈ A ∣ φ} = {x ∈ B ∣ φ})

Syntax hints:   → wi 4   = wceq 1642  {crab 2618

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623

This theorem is referenced by:  rabeqbidv  2854  rabeqbidva  2855  ifeq1  3666  ifeq2  3667