Theorem cbvabv 2356

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 1576	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1576	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvab 2355	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397  {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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224

This theorem is referenced by:  cdeqab1  3023  difjust  3201  unjust  3203  injust  3205  uniiunlem  3316  dfif3  3619  pwjust  3653  snjust  3674  intab  3957  iotajust  5285  cbviotavw  5292  tfrlemi1  6498  tfr1onlemaccex  6514  tfrcllemaccex  6527  frecsuc  6573  isbth  7166  nqprlu  7767  recexpr  7858  caucvgprprlemval  7908  caucvgprprlemnbj  7913  caucvgprprlemaddq  7928  caucvgprprlem1  7929  caucvgprprlem2  7930  axcaucvg  8120  mertensabs  12103  4sq  12988  isuhgrm  15928  isushgrm  15929  isupgren  15952  isumgren  15962  isuspgren  16014  isusgren  16015  bds  16472