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Theorem bj-elequ2g 31687
Description: A form of elequ2 1952 with a universal quantifier. Its converse is ax-ext 2494. (TODO: move to main part, minimize axext4 2498--- as of 4-Nov-2020, minimizes only axext4 2498, by 13 bytes; and link to it in the comment of ax-ext 2494.) (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 1952 . 2 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
21alrimiv 1808 1 (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by:  bj-axext4  31799  bj-cleqhyp  31915
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