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  3020  difjust  3198  unjust  3200  injust  3202  uniiunlem  3313  dfif3  3616  pwjust  3650  snjust  3671  intab  3952  iotajust  5277  cbviotavw  5284  tfrlemi1  6484  tfr1onlemaccex  6500  tfrcllemaccex  6513  frecsuc  6559  isbth  7145  nqprlu  7745  recexpr  7836  caucvgprprlemval  7886  caucvgprprlemnbj  7891  caucvgprprlemaddq  7906  caucvgprprlem1  7907  caucvgprprlem2  7908  axcaucvg  8098  mertensabs  12063  4sq  12948  isuhgrm  15886  isushgrm  15887  isupgren  15910  isumgren  15920  isuspgren  15970  isusgren  15971  bds  16269