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Theorem cbv2 1676
Description: Rule used to change bound variables, using implicit substitution. Revised to align format of 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
cbv2.1 𝑥𝜑
cbv2.2 𝑦𝜑
cbv2.3 (𝜑 → Ⅎ𝑦𝜓)
cbv2.4 (𝜑 → Ⅎ𝑥𝜒)
cbv2.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbv2 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Proof of Theorem cbv2
StepHypRef Expression
1 cbv2.2 . . . 4 𝑦𝜑
21nfri 1453 . . 3 (𝜑 → ∀𝑦𝜑)
3 cbv2.1 . . . . 5 𝑥𝜑
43nfal 1509 . . . 4 𝑥𝑦𝜑
54nfri 1453 . . 3 (∀𝑦𝜑 → ∀𝑥𝑦𝜑)
62, 5syl 14 . 2 (𝜑 → ∀𝑥𝑦𝜑)
7 cbv2.3 . . . 4 (𝜑 → Ⅎ𝑦𝜓)
87nfrd 1454 . . 3 (𝜑 → (𝜓 → ∀𝑦𝜓))
9 cbv2.4 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
109nfrd 1454 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
11 cbv2.5 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
128, 10, 11cbv2h 1675 . 2 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
136, 12syl 14 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  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-8 1436  ax-4 1441  ax-17 1460  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:  cbvald  1842  cbvrald  10749
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