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Theorem cbvalh 1709
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 143 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 4cbv3h 1704 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
63equcoms 1667 . . . 4 (𝑦 = 𝑥 → (𝜑𝜓))
76biimprd 157 . . 3 (𝑦 = 𝑥 → (𝜓𝜑))
82, 1, 7cbv3h 1704 . 2 (∀𝑦𝜓 → ∀𝑥𝜑)
95, 8impbii 125 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1312
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-nf 1420
This theorem is referenced by:  cbval  1710  sb8h  1808  cbvalv  1869  sb9v  1929  sb8euh  1998
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