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Theorem ralbii2 2450
 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 1447 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵𝜓))
3 df-ral 2423 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
4 df-ral 2423 . 2 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
52, 3, 43bitr4i 211 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330   ∈ wcel 2112  ∀wral 2418 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426 This theorem depends on definitions:  df-bi 116  df-ral 2423 This theorem is referenced by:  raleqbii  2452  ralbiia  2454  ralrab  2851  raldifb  3223  raluz2  9430  ralrp  9521  isprm4  11872
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