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Theorem cbvalh 1764
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 144 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 4cbv3h 1754 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
63equcoms 1719 . . . 4 (𝑦 = 𝑥 → (𝜑𝜓))
76biimprd 158 . . 3 (𝑦 = 𝑥 → (𝜓𝜑))
82, 1, 7cbv3h 1754 . 2 (∀𝑦𝜓 → ∀𝑥𝜑)
95, 8impbii 126 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1362
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-nf 1472
This theorem is referenced by:  cbval  1765  sb8h  1865  cbvalv  1929  sb9v  1994  sb8euh  2065
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