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Theorem cbv1v 1734
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 16-Jun-2019.)
Hypotheses
Ref Expression
cbv1v.1 𝑥𝜑
cbv1v.2 𝑦𝜑
cbv1v.3 (𝜑 → Ⅎ𝑦𝜓)
cbv1v.4 (𝜑 → Ⅎ𝑥𝜒)
cbv1v.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbv1v (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem cbv1v
StepHypRef Expression
1 cbv1v.2 . . . . 5 𝑦𝜑
2 cbv1v.3 . . . . 5 (𝜑 → Ⅎ𝑦𝜓)
31, 2nfim1 1558 . . . 4 𝑦(𝜑𝜓)
4 cbv1v.1 . . . . 5 𝑥𝜑
5 cbv1v.4 . . . . 5 (𝜑 → Ⅎ𝑥𝜒)
64, 5nfim1 1558 . . . 4 𝑥(𝜑𝜒)
7 cbv1v.5 . . . . . 6 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
87com12 30 . . . . 5 (𝑥 = 𝑦 → (𝜑 → (𝜓𝜒)))
98a2d 26 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) → (𝜑𝜒)))
103, 6, 9cbv3v 1731 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑦(𝜑𝜒))
11419.21 1570 . . 3 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
12119.21 1570 . . 3 (∀𝑦(𝜑𝜒) ↔ (𝜑 → ∀𝑦𝜒))
1310, 11, 123imtr3i 199 . 2 ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑦𝜒))
1413pm2.86i 98 1 (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1340  wnf 1447
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-i9 1517  ax-ial 1521  ax-i5r 1522
This theorem depends on definitions:  df-bi 116  df-nf 1448
This theorem is referenced by:  cbv2w  1737
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