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Theorem cbvalv 1839
Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
cbvalv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvalv (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvalv
StepHypRef Expression
1 ax-17 1462 . 2 (𝜑 → ∀𝑦𝜑)
2 ax-17 1462 . 2 (𝜓 → ∀𝑥𝜓)
3 cbvalv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvalh 1680 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by:  nfcjust  2213  cdeqal1  2820  dfss4st  3221  zfpow  3985  tfisi  4375  acexmid  5612  tfrlem3-2d  6031  tfrlemi1  6051  tfrexlem  6053  tfr1onlemaccex  6067  tfrcllemaccex  6080  findcard  6556  fisseneq  6592  genprndl  7024  genprndu  7025  zfz1iso  10143
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