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Theorem raleqbidva 3330
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 3174 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
3 raleqbidva.1 . . 3 (𝜑𝐴 = 𝐵)
43raleqdv 3324 . 2 (𝜑 → (∀𝑥𝐴 𝜒 ↔ ∀𝑥𝐵 𝜒))
52, 4bitrd 279 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-ral 3060  df-rex 3069
This theorem is referenced by:  raleqbidvv  3332  catpropd  17754  cidpropd  17755  funcpropd  17954  fullpropd  17974  natpropd  18033  gsumpropd2lem  18705  ringurd  20203  istrkgcb  28479  iscgrg  28535  isperp  28735  clwlkclwwlk  30031  urpropd  33222  lindfpropd  33390  opprqus0g  33498  opprqusdrng  33501  ist0cld  33794  matunitlindflem1  37603  primrootsunit1  42079  sticksstones3  42130
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