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  3951  iotajust  5276  cbviotavw  5283  tfrlemi1  6476  tfr1onlemaccex  6492  tfrcllemaccex  6505  frecsuc  6551  isbth  7130  nqprlu  7730  recexpr  7821  caucvgprprlemval  7871  caucvgprprlemnbj  7876  caucvgprprlemaddq  7891  caucvgprprlem1  7892  caucvgprprlem2  7893  axcaucvg  8083  mertensabs  12043  4sq  12928  isuhgrm  15865  isushgrm  15866  isupgren  15889  isumgren  15899  isuspgren  15949  isusgren  15950  bds  16172