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

Theorem raleqbidva 3430
 Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
Hypotheses
Ref Expression
raleqbidva.1 (𝜑𝐴 = 𝐵)
raleqbidva.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
raleqbidva (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem raleqbidva
StepHypRef Expression
1 raleqbidva.2 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
21ralbidva 3200 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
3 raleqbidva.1 . . 3 (𝜑𝐴 = 𝐵)
43raleqdv 3420 . 2 (𝜑 → (∀𝑥𝐴 𝜒 ↔ ∀𝑥𝐵 𝜒))
52, 4bitrd 280 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1530   ∈ wcel 2106  ∀wral 3142 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-ext 2796 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-cleq 2817  df-clel 2897  df-ral 3147 This theorem is referenced by:  catpropd  16971  cidpropd  16972  funcpropd  17162  fullpropd  17182  natpropd  17238  gsumpropd2lem  17880  istrkgc  26154  istrkgb  26155  istrkgcb  26156  istrkge  26157  iscgrg  26212  isperp  26412  clwlkclwwlk  27695  rngurd  30772  lindfpropd  30857  matunitlindflem1  34756
 Copyright terms: Public domain W3C validator