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Theorem raleqbidva 3325
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 3173 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
3 raleqbidva.1 . . 3 (𝜑𝐴 = 𝐵)
43raleqdv 3323 . 2 (𝜑 → (∀𝑥𝐴 𝜒 ↔ ∀𝑥𝐵 𝜒))
52, 4bitrd 278 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-cleq 2722  df-ral 3060  df-rex 3069
This theorem is referenced by:  raleqbidvv  3327  catpropd  17657  cidpropd  17658  funcpropd  17855  fullpropd  17875  natpropd  17933  gsumpropd2lem  18604  ringurd  20079  istrkgcb  27974  iscgrg  28030  isperp  28230  clwlkclwwlk  29522  urpropd  32648  lindfpropd  32772  opprqus0g  32878  opprqusdrng  32881  ist0cld  33111  matunitlindflem1  36787  sticksstones3  41270  sticksstones11  41278
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