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  3654  pwjust  3689  snjust  3713  intab  3997  iotajust  5334  cbviotavw  5341  tfrlemi1  6596  tfr1onlemaccex  6612  tfrcllemaccex  6625  frecsuc  6671  isbth  7277  nqprlu  7907  recexpr  7998  caucvgprprlemval  8048  caucvgprprlemnbj  8053  caucvgprprlemaddq  8068  caucvgprprlem1  8069  caucvgprprlem2  8070  axcaucvg  8260  hashf1lem2  11267  mertensabs  12285  4sq  13170  ballotfilemfmpn  13215  isuhgrm  16229  isushgrm  16230  isupgren  16253  isumgren  16263  isuspgren  16315  isusgren  16316  bds  16794