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Theorem cbv1 1723
Description: Rule used to change bound variables, using implicit substitution. Revised to format hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
Hypotheses
Ref Expression
cbv1.1 𝑥𝜑
cbv1.2 𝑦𝜑
cbv1.3 (𝜑 → Ⅎ𝑦𝜓)
cbv1.4 (𝜑 → Ⅎ𝑥𝜒)
cbv1.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbv1 (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Proof of Theorem cbv1
StepHypRef Expression
1 cbv1.2 . . . . 5 𝑦𝜑
2 cbv1.3 . . . . 5 (𝜑 → Ⅎ𝑦𝜓)
31, 2nfim1 1551 . . . 4 𝑦(𝜑𝜓)
4 cbv1.1 . . . . 5 𝑥𝜑
5 cbv1.4 . . . . 5 (𝜑 → Ⅎ𝑥𝜒)
64, 5nfim1 1551 . . . 4 𝑥(𝜑𝜒)
7 cbv1.5 . . . . . 6 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
87com12 30 . . . . 5 (𝑥 = 𝑦 → (𝜑 → (𝜓𝜒)))
98a2d 26 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) → (𝜑𝜒)))
103, 6, 9cbv3 1721 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑦(𝜑𝜒))
11419.21 1563 . . 3 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
12119.21 1563 . . 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 1330  wnf 1437
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-i9 1511  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116  df-nf 1438
This theorem is referenced by:  cbv1h  1724
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