Theorem rabeq 2673

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 2279	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2279	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	rabeqf 2671	1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1331  {crab 2418

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423

This theorem is referenced by:  rabeqdv  2675  rabeqbidv  2676  rabeqbidva  2677  difeq1  3182  ifeq1  3472  ifeq2  3473  elfvmptrab  5509  pmvalg  6546  unfiexmid  6799  ssfirab  6815  supeq2  6869  iooval2  9691  fzval2  9786  lcmval  11733  lcmcllem  11737  lcmledvds  11740  clsfval  12259