Theorem cbvabv 2365

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 1581	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1581	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvab 2364	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402  {cab 2224

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231

This theorem is referenced by:  eqabbw  2375  cdeqab1  3043  difjust  3221  unjust  3223  injust  3225  uniiunlem  3338  dfif3  3651  pwjust  3686  snjust  3710  intab  3994  iotajust  5331  cbviotavw  5338  tfrlemi1  6593  tfr1onlemaccex  6609  tfrcllemaccex  6622  frecsuc  6668  isbth  7274  nqprlu  7904  recexpr  7995  caucvgprprlemval  8045  caucvgprprlemnbj  8050  caucvgprprlemaddq  8065  caucvgprprlem1  8066  caucvgprprlem2  8067  axcaucvg  8257  hashf1lem2  11264  mertensabs  12282  4sq  13167  ballotfilemfmpn  13212  isuhgrm  16226  isushgrm  16227  isupgren  16250  isumgren  16260  isuspgren  16312  isusgren  16313  bds  16791