Theorem rabeq 2744

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 2332	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2332	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	rabeqf 2742	1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1364  {crab 2472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477

This theorem is referenced by:  rabeqdv  2746  rabeqbidv  2747  rabeqbidva  2748  difeq1  3261  ifeq1  3552  ifeq2  3553  elfvmptrab  5632  pmvalg  6686  unfiexmid  6947  ssfirab  6963  supeq2  7019  iooval2  9947  fzval2  10043  clsfval  14078