Theorem cbvrabv 2814

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 2386	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2386	. 2 ⊢ Ⅎ𝑦𝐴
3		nfv 1577	. 2 ⊢ Ⅎ𝑦𝜑
4		nfv 1577	. 2 ⊢ Ⅎ𝑥𝜓
5		cbvrabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvrab 2813	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  {crab 2526

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531

This theorem is referenced by:  pwnss  4277  acexmidlemv  6056  exmidac  7529  genipv  7840  ltexpri  7944  suplocsrlempr  8138  suplocsr  8140  zsupssdc  10622  hashfibc  11232  bitsfzolem  12665  nninfctlemfo  12761  sqne2sq  12899  eulerth  12955  odzval  12964  pcprecl  13012  pcprendvds  13013  pcpremul  13016  pceulem  13017  4sqlem19  13132  ballotfilemelo  13166  ballotfileme  13180  ballotfilemimin  13193  ballotfilemfrcn0  13217  ballotfilem7  13223  ballotfi  13226  lfgredg2dom  16253  vtxdumgrfival  16419  vtxduspgrfvedgfilem  16421  vtxduspgrfvedgfi  16422