Theorem cbvrabv 2798

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)

Hypothesis

Ref	Expression
cbvrabv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvrabv	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}

Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvrabv

Step	Hyp	Ref	Expression
1		nfcv 2372	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2372	. 2 ⊢ Ⅎ𝑦𝐴
3		nfv 1574	. 2 ⊢ Ⅎ𝑦𝜑
4		nfv 1574	. 2 ⊢ Ⅎ𝑥𝜓
5		cbvrabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvrab 2797	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395  {crab 2512

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517

This theorem is referenced by:  pwnss  4242  acexmidlemv  5998  exmidac  7387  genipv  7692  ltexpri  7796  suplocsrlempr  7990  suplocsr  7992  zsupssdc  10453  bitsfzolem  12460  nninfctlemfo  12556  sqne2sq  12694  eulerth  12750  odzval  12759  pcprecl  12807  pcprendvds  12808  pcpremul  12811  pceulem  12812  4sqlem19  12927  lfgredg2dom  15924