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Theorem cbval 1747
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
cbval.1 𝑦𝜑
cbval.2 𝑥𝜓
cbval.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbval (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Proof of Theorem cbval
StepHypRef Expression
1 cbval.1 . . 3 𝑦𝜑
21nfri 1512 . 2 (𝜑 → ∀𝑦𝜑)
3 cbval.2 . . 3 𝑥𝜓
43nfri 1512 . 2 (𝜓 → ∀𝑥𝜓)
5 cbval.3 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
62, 4, 5cbvalh 1746 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1346  wnf 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-nf 1454
This theorem is referenced by:  sb8  1849  cbval2  1914  sb8eu  2032  abbi  2284  cleqf  2337  cbvralf  2689  ralab2  2894  cbvralcsf  3111  dfss2f  3138  elintab  3840  cbviota  5163  sb8iota  5165  dffun6f  5209  dffun4f  5212  mptfvex  5579  findcard2  6865  findcard2s  6866
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