Theorem rabeq 2718

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 2308	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2308	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	rabeqf 2716	1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1343  {crab 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453

This theorem is referenced by:  rabeqdv  2720  rabeqbidv  2721  rabeqbidva  2722  difeq1  3233  ifeq1  3523  ifeq2  3524  elfvmptrab  5581  pmvalg  6625  unfiexmid  6883  ssfirab  6899  supeq2  6954  iooval2  9851  fzval2  9947  lcmval  11995  lcmcllem  11999  lcmledvds  12002  clsfval  12741