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

Theorem cbval 2400
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. Check out cbvalw 2031, cbvalvw 2032, cbvalv1 2341 for versions requiring fewer axioms. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbval.1 𝑦𝜑
cbval.2 𝑥𝜓
cbval.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbval (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Proof of Theorem cbval
StepHypRef Expression
1 cbval.1 . . 3 𝑦𝜑
2 cbval.2 . . 3 𝑥𝜓
3 cbval.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43biimpd 229 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 4cbv3 2399 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
63biimprd 248 . . . 4 (𝑥 = 𝑦 → (𝜓𝜑))
76equcoms 2016 . . 3 (𝑦 = 𝑥 → (𝜓𝜑))
82, 1, 7cbv3 2399 . 2 (∀𝑦𝜓 → ∀𝑥𝜑)
95, 8impbii 209 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1534  wnf 1779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-11 2154  ax-12 2174  ax-13 2374
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-nf 1780
This theorem is referenced by:  cbvex  2401  cbvalv  2402  cbval2  2413  sb8  2519
  Copyright terms: Public domain W3C validator