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Theorem cbv2h 2414
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 11-May-1993.)
Hypotheses
Ref Expression
cbv2h.1 (𝜑 → (𝜓 → ∀𝑦𝜓))
cbv2h.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
cbv2h.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbv2h (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Proof of Theorem cbv2h
StepHypRef Expression
1 cbv2h.1 . . 3 (𝜑 → (𝜓 → ∀𝑦𝜓))
2 cbv2h.2 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
3 cbv2h.3 . . . 4 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
4 biimp 205 . . . 4 ((𝜓𝜒) → (𝜓𝜒))
53, 4syl6 35 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
61, 2, 5cbv1h 2413 . 2 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
7 equcomi 2099 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
8 biimpr 210 . . . . 5 ((𝜓𝜒) → (𝜒𝜓))
97, 3, 8syl56 36 . . . 4 (𝜑 → (𝑦 = 𝑥 → (𝜒𝜓)))
102, 1, 9cbv1h 2413 . . 3 (∀𝑦𝑥𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
1110alcoms 2184 . 2 (∀𝑥𝑦𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
126, 11impbid 202 1 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859
This theorem is referenced by:  cbv2  2415  eujustALT  2610
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