Theorem cbvabv 2295

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 1521	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1521	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvab 2294	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348  {cab 2156

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163

This theorem is referenced by:  cdeqab1  2947  difjust  3122  unjust  3124  injust  3126  uniiunlem  3236  dfif3  3539  pwjust  3567  snjust  3588  intab  3860  iotajust  5159  tfrlemi1  6311  tfr1onlemaccex  6327  tfrcllemaccex  6340  frecsuc  6386  isbth  6944  nqprlu  7509  recexpr  7600  caucvgprprlemval  7650  caucvgprprlemnbj  7655  caucvgprprlemaddq  7670  caucvgprprlem1  7671  caucvgprprlem2  7672  axcaucvg  7862  mertensabs  11500  bds  13886