Theorem cbvabv 2330

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 1551	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1551	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvabv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvab 2329	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1373  {cab 2191

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198

This theorem is referenced by:  cdeqab1  2990  difjust  3167  unjust  3169  injust  3171  uniiunlem  3282  dfif3  3584  pwjust  3617  snjust  3638  intab  3914  iotajust  5231  tfrlemi1  6418  tfr1onlemaccex  6434  tfrcllemaccex  6447  frecsuc  6493  isbth  7069  nqprlu  7660  recexpr  7751  caucvgprprlemval  7801  caucvgprprlemnbj  7806  caucvgprprlemaddq  7821  caucvgprprlem1  7822  caucvgprprlem2  7823  axcaucvg  8013  mertensabs  11848  4sq  12733  bds  15787