Theorem cbvabv 2357

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)

Hypothesis

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

Assertion

Ref	Expression
cbvabv	⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥

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

Proof of Theorem cbvabv

Step	Hyp	Ref	Expression
1		nfv 1577	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1577	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvab 2356	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  {cab 2217

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 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224

This theorem is referenced by:  cdeqab1  3024  difjust  3202  unjust  3204  injust  3206  uniiunlem  3318  dfif3  3623  pwjust  3657  snjust  3678  intab  3962  iotajust  5292  cbviotavw  5299  tfrlemi1  6541  tfr1onlemaccex  6557  tfrcllemaccex  6570  frecsuc  6616  isbth  7209  nqprlu  7810  recexpr  7901  caucvgprprlemval  7951  caucvgprprlemnbj  7956  caucvgprprlemaddq  7971  caucvgprprlem1  7972  caucvgprprlem2  7973  axcaucvg  8163  mertensabs  12161  4sq  13046  isuhgrm  15995  isushgrm  15996  isupgren  16019  isumgren  16029  isuspgren  16081  isusgren  16082  bds  16550