Theorem cbvabv 2282

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 1508	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1508	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvab 2281	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1335  {cab 2143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150

This theorem is referenced by:  cdeqab1  2929  difjust  3103  unjust  3105  injust  3107  uniiunlem  3216  dfif3  3518  pwjust  3544  snjust  3565  intab  3836  iotajust  5131  tfrlemi1  6273  tfr1onlemaccex  6289  tfrcllemaccex  6302  frecsuc  6348  isbth  6904  nqprlu  7450  recexpr  7541  caucvgprprlemval  7591  caucvgprprlemnbj  7596  caucvgprprlemaddq  7611  caucvgprprlem1  7612  caucvgprprlem2  7613  axcaucvg  7803  mertensabs  11416  bds  13385