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Theorem cbv1 1673
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 1504 . . . 4 𝑦(𝜑𝜓)
4 cbv1.1 . . . . 5 𝑥𝜑
5 cbv1.4 . . . . 5 (𝜑 → Ⅎ𝑥𝜒)
64, 5nfim1 1504 . . . 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 1671 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑦(𝜑𝜒))
11419.21 1516 . . 3 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
12119.21 1516 . . 3 (∀𝑦(𝜑𝜒) ↔ (𝜑 → ∀𝑦𝜒))
1310, 11, 123imtr3i 198 . 2 ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑦𝜒))
1413pm2.86i 97 1 (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wnf 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  cbv1h  1674
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