Theorem cbvabv 2354

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 1574	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1574	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvab 2353	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395  {cab 2215

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222

This theorem is referenced by:  cdeqab1  3021  difjust  3199  unjust  3201  injust  3203  uniiunlem  3314  dfif3  3617  pwjust  3651  snjust  3672  intab  3955  iotajust  5283  cbviotavw  5290  tfrlemi1  6493  tfr1onlemaccex  6509  tfrcllemaccex  6522  frecsuc  6568  isbth  7157  nqprlu  7757  recexpr  7848  caucvgprprlemval  7898  caucvgprprlemnbj  7903  caucvgprprlemaddq  7918  caucvgprprlem1  7919  caucvgprprlem2  7920  axcaucvg  8110  mertensabs  12088  4sq  12973  isuhgrm  15912  isushgrm  15913  isupgren  15936  isumgren  15946  isuspgren  15996  isusgren  15997  bds  16382