Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbv2w Structured version   Visualization version   GIF version

Theorem cbv2w 2357
 Description: Rule used to change bound variables, using implicit substitution. Version of cbv2 2423 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 5-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbv2w.1 𝑥𝜑
cbv2w.2 𝑦𝜑
cbv2w.3 (𝜑 → Ⅎ𝑦𝜓)
cbv2w.4 (𝜑 → Ⅎ𝑥𝜒)
cbv2w.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbv2w (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem cbv2w
StepHypRef Expression
1 cbv2w.1 . . 3 𝑥𝜑
2 cbv2w.2 . . 3 𝑦𝜑
3 cbv2w.3 . . 3 (𝜑 → Ⅎ𝑦𝜓)
4 cbv2w.4 . . 3 (𝜑 → Ⅎ𝑥𝜒)
5 cbv2w.5 . . . 4 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
6 biimp 217 . . . 4 ((𝜓𝜒) → (𝜓𝜒))
75, 6syl6 35 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
81, 2, 3, 4, 7cbv1v 2356 . 2 (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
9 equcomi 2024 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
10 biimpr 222 . . . 4 ((𝜓𝜒) → (𝜒𝜓))
119, 5, 10syl56 36 . . 3 (𝜑 → (𝑦 = 𝑥 → (𝜒𝜓)))
122, 1, 4, 3, 11cbv1v 2356 . 2 (𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
138, 12impbid 214 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535  Ⅎwnf 1784 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-11 2161  ax-12 2177 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785 This theorem is referenced by:  cbvaldw  2358  cbval2v  2363  cbveud  34670
 Copyright terms: Public domain W3C validator