Theorem cbvabv 2332

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 1552	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1552	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvab 2331	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1373  {cab 2193

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200

This theorem is referenced by:  cdeqab1  2997  difjust  3175  unjust  3177  injust  3179  uniiunlem  3290  dfif3  3593  pwjust  3627  snjust  3648  intab  3928  iotajust  5250  tfrlemi1  6441  tfr1onlemaccex  6457  tfrcllemaccex  6470  frecsuc  6516  isbth  7095  nqprlu  7695  recexpr  7786  caucvgprprlemval  7836  caucvgprprlemnbj  7841  caucvgprprlemaddq  7856  caucvgprprlem1  7857  caucvgprprlem2  7858  axcaucvg  8048  mertensabs  11963  4sq  12848  isuhgrm  15782  isushgrm  15783  isupgren  15806  isumgren  15816  bds  15986