Theorem cbvabv 2361

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 2360	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  {cab 2220

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

This theorem is referenced by:  cdeqab1  3037  difjust  3215  unjust  3217  injust  3219  uniiunlem  3332  dfif3  3640  pwjust  3675  snjust  3699  intab  3983  iotajust  5316  cbviotavw  5323  tfrlemi1  6576  tfr1onlemaccex  6592  tfrcllemaccex  6605  frecsuc  6651  isbth  7250  nqprlu  7878  recexpr  7969  caucvgprprlemval  8019  caucvgprprlemnbj  8024  caucvgprprlemaddq  8039  caucvgprprlem1  8040  caucvgprprlem2  8041  axcaucvg  8231  mertensabs  12248  4sq  13133  ballotfilemfmpn  13178  isuhgrm  16192  isushgrm  16193  isupgren  16216  isumgren  16226  isuspgren  16278  isusgren  16279  bds  16747