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

Theorem cbvaldvaw 2045
 Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva 2419 with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017.) (Revised by Gino Giotto, 10-Jan-2024.) Reduce axiom usage, along an idea of Gino Giotto. (Revised by Wolf Lammen, 10-Feb-2024.)
Hypothesis
Ref Expression
cbvaldvaw.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
cbvaldvaw (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvaldvaw
StepHypRef Expression
1 cbvaldvaw.1 . . . . . 6 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
21ancoms 462 . . . . 5 ((𝑥 = 𝑦𝜑) → (𝜓𝜒))
32pm5.74da 803 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
43cbvalvw 2043 . . 3 (∀𝑥(𝜑𝜓) ↔ ∀𝑦(𝜑𝜒))
5 19.21v 1940 . . 3 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
6 19.21v 1940 . . 3 (∀𝑦(𝜑𝜒) ↔ (𝜑 → ∀𝑦𝜒))
74, 5, 63bitr3i 304 . 2 ((𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑦𝜒))
87pm5.74ri 275 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by:  cbvexdvaw  2046  cbval2vw  2047  scottabf  41356  ismnu  41377
 Copyright terms: Public domain W3C validator