Theorem cbvabv 2329

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 1550	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1550	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvab 2328	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1372  {cab 2190

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197

This theorem is referenced by:  cdeqab1  2989  difjust  3166  unjust  3168  injust  3170  uniiunlem  3281  dfif3  3583  pwjust  3616  snjust  3637  intab  3913  iotajust  5230  tfrlemi1  6417  tfr1onlemaccex  6433  tfrcllemaccex  6446  frecsuc  6492  isbth  7068  nqprlu  7659  recexpr  7750  caucvgprprlemval  7800  caucvgprprlemnbj  7805  caucvgprprlemaddq  7820  caucvgprprlem1  7821  caucvgprprlem2  7822  axcaucvg  8012  mertensabs  11819  4sq  12704  bds  15749