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Theorem cbv2h 2257
 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 204 . . . 4 ((𝜓𝜒) → (𝜓𝜒))
53, 4syl6 34 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
61, 2, 5cbv1h 2256 . 2 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
7 equcomi 1931 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
8 biimpr 209 . . . . 5 ((𝜓𝜒) → (𝜒𝜓))
97, 3, 8syl56 35 . . . 4 (𝜑 → (𝑦 = 𝑥 → (𝜒𝜓)))
102, 1, 9cbv1h 2256 . . 3 (∀𝑦𝑥𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
1110alcoms 2022 . 2 (∀𝑥𝑦𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
126, 11impbid 201 1 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701 This theorem is referenced by:  cbv2  2258  eujustALT  2461
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