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Theorem cbvalh 1636
 Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
cbvalh.1 (𝜑 → ∀𝑦𝜑)
cbvalh.2 (𝜓 → ∀𝑥𝜓)
cbvalh.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvalh (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Proof of Theorem cbvalh
StepHypRef Expression
1 cbvalh.1 . . 3 (𝜑 → ∀𝑦𝜑)
2 cbvalh.2 . . 3 (𝜓 → ∀𝑥𝜓)
3 cbvalh.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43biimpd 132 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 4cbv3h 1631 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
63equcoms 1594 . . . 4 (𝑦 = 𝑥 → (𝜑𝜓))
76biimprd 147 . . 3 (𝑦 = 𝑥 → (𝜓𝜑))
82, 1, 7cbv3h 1631 . 2 (∀𝑦𝜓 → ∀𝑥𝜑)
95, 8impbii 117 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350 This theorem is referenced by:  cbval  1637  sb8h  1734  cbvalv  1794  sb9v  1854  sb8euh  1923
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