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Theorem elequ2g 2096
Description: A form of elequ2 2094 with a universal quantifier. Its converse is the axiom of extensionality ax-ext 2767. (Contributed by BJ, 3-Oct-2019.)
Assertion
Ref Expression
elequ2g (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem elequ2g
StepHypRef Expression
1 elequ2 2094 . 2 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
21alrimiv 1903 1 (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-9 2089
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1760
This theorem is referenced by:  axextb  2770
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