Theorem rabeq 2614

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 2229	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2229	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	rabeqf 2612	1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1290  {crab 2364

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rab 2369

This theorem is referenced by:  rabeqdv  2616  rabeqbidv  2617  rabeqbidva  2618  difeq1  3114  ifeq1  3402  ifeq2  3403  pmvalg  6432  unfiexmid  6684  ssfirab  6699  supeq2  6740  iooval2  9396  fzval2  9490  lcmval  11386  lcmcllem  11390  lcmledvds  11393  clsfval  11864