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Theorem bj-elequ2g 33003
Description: A form of elequ2 2159 with a universal quantifier. Its converse is ax-ext 2751. (TODO: move to main part, minimize axext4 2755--- as of 4-Nov-2020, minimizes only axext4 2755, by 13 bytes; and link to it in the comment of ax-ext 2751.) (Contributed by BJ, 3-Oct-2019.)
Assertion
Ref Expression
bj-elequ2g (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem bj-elequ2g
StepHypRef Expression
1 elequ2 2159 . 2 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
21alrimiv 2007 1 (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853
This theorem is referenced by:  bj-axext4  33106  bj-cleqhyp  33221
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