MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbv2h Structured version   Visualization version   GIF version

Theorem cbv2h 2268
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 2267 . 2 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
7 equcomi 1943 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
8 biimpr 210 . . . . 5 ((𝜓𝜒) → (𝜒𝜓))
97, 3, 8syl56 36 . . . 4 (𝜑 → (𝑦 = 𝑥 → (𝜒𝜓)))
102, 1, 9cbv1h 2267 . . 3 (∀𝑦𝑥𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
1110alcoms 2034 . 2 (∀𝑥𝑦𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
126, 11impbid 202 1 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1704  df-nf 1709
This theorem is referenced by:  cbv2  2269  eujustALT  2472
  Copyright terms: Public domain W3C validator