Theorem cbvrabv 2729

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 2312	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2312	. 2 ⊢ Ⅎ𝑦𝐴
3		nfv 1521	. 2 ⊢ Ⅎ𝑦𝜑
4		nfv 1521	. 2 ⊢ Ⅎ𝑥𝜓
5		cbvrabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvrab 2728	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348  {crab 2452

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457

This theorem is referenced by:  pwnss  4145  acexmidlemv  5851  exmidac  7186  genipv  7471  ltexpri  7575  suplocsrlempr  7769  suplocsr  7771  zsupssdc  11909  sqne2sq  12131  eulerth  12187  odzval  12195  pcprecl  12243  pcprendvds  12244  pcpremul  12247  pceulem  12248