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Theorem cbvalv 1810
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 1435 . 2 (𝜑 → ∀𝑦𝜑)
2 ax-17 1435 . 2 (𝜓 → ∀𝑥𝜓)
3 cbvalv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvalh 1652 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  nfcjust  2182  cdeqal1  2777  zfpow  3955  tfisi  4337  acexmid  5538  tfrlem3-2d  5958  tfrlemi1  5976  tfrexlem  5978  findcard  6375  genprndl  6676  genprndu  6677
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