Theorem cbvrabv 2820

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 2392	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2392	. 2 ⊢ Ⅎ𝑦𝐴
3		nfv 1581	. 2 ⊢ Ⅎ𝑦𝜑
4		nfv 1581	. 2 ⊢ Ⅎ𝑥𝜓
5		cbvrabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvrab 2819	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   = wceq 1402  {crab 2532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537

This theorem is used by:  pwnss  4296  acexmidlemv  6083  exmidac  7565  genipv  7876  ltexpri  7980  suplocsrlempr  8174  suplocsr  8176  zsupssdc  10673  hashfibc  11283  bitsfzolem  12721  nninfctlemfo  12817  sqne2sq  12955  eulerth  13011  odzval  13020  pcprecl  13068  pcprendvds  13069  pcpremul  13072  pceulem  13073  4sqlem19  13188  ballotfilemelo  13222  ballotfileme  13236  ballotfilemimin  13249  ballotfilemfrcn0  13273  ballotfilem7  13279  ballotfi  13282  lfgredg2dom  16373  vtxdumgrfival  16539  vtxduspgrfvedgfilem  16541  vtxduspgrfvedgfi  16542