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Theorem cbvalv 1917
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 1526 . 2 (𝜑 → ∀𝑦𝜑)
2 ax-17 1526 . 2 (𝜓 → ∀𝑥𝜓)
3 cbvalv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvalh 1753 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351
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-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-nf 1461
This theorem is referenced by:  nfcjust  2307  cdeqal1  2953  dfss4st  3368  zfpow  4174  tfisi  4585  acexmid  5871  tfrlem3-2d  6310  tfrlemi1  6330  tfrexlem  6332  tfr1onlemaccex  6346  tfrcllemaccex  6359  findcard  6885  fisseneq  6928  genprndl  7517  genprndu  7518  zfz1iso  10814
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