Theorem cbvabv 2359

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 1577	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1577	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvab 2358	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  {cab 2218

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225

This theorem is referenced by:  cdeqab1  3034  difjust  3212  unjust  3214  injust  3216  uniiunlem  3328  dfif3  3636  pwjust  3670  snjust  3694  intab  3978  iotajust  5311  cbviotavw  5318  tfrlemi1  6563  tfr1onlemaccex  6579  tfrcllemaccex  6592  frecsuc  6638  isbth  7237  nqprlu  7862  recexpr  7953  caucvgprprlemval  8003  caucvgprprlemnbj  8008  caucvgprprlemaddq  8023  caucvgprprlem1  8024  caucvgprprlem2  8025  axcaucvg  8215  mertensabs  12223  4sq  13108  isuhgrm  16066  isushgrm  16067  isupgren  16090  isumgren  16100  isuspgren  16152  isusgren  16153  bds  16621