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Theorem ralbii2 2334
Description: Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)
Hypothesis
Ref Expression
ralbii2.1 ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜓))
Assertion
Ref Expression
ralbii2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓)

Proof of Theorem ralbii2
StepHypRef Expression
1 ralbii2.1 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜓))
21albii 1359 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵𝜓))
3 df-ral 2311 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
4 df-ral 2311 . 2 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
52, 3, 43bitr4i 201 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1241  wcel 1393  wral 2306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338
This theorem depends on definitions:  df-bi 110  df-ral 2311
This theorem is referenced by:  raleqbii  2336  ralbiia  2338  ralrab  2702  raldifb  3083  raluz2  8522  ralrp  8604
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