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Theorem raleleq 3400
Description: All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020.)
Assertion
Ref Expression
raleleq (𝐴 = 𝐵 → ∀𝑥𝐴 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem raleleq
StepHypRef Expression
1 eleq2 2902 . . 3 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21biimpd 232 . 2 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
32ralrimiv 3173 1 (𝐴 = 𝐵 → ∀𝑥𝐴 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  wral 3130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2815  df-clel 2894  df-ral 3135
This theorem is referenced by:  uvtxnbgrb  27189
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