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Theorem cleljustALT 2135
Description: Alternate proof of cleljust 1945. It is kept here and should not be modified because it is referenced on the Metamath Proof Explorer Home Page (mmset.html) as an example of how DV conditions are inherited by substitutions. (Contributed by NM, 28-Jan-2004.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cleljustALT (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem cleljustALT
StepHypRef Expression
1 ax-5 1789 . . 3 (𝑥𝑦 → ∀𝑧 𝑥𝑦)
2 elequ1 1944 . . 3 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
31, 2equsexhv 2119 . 2 (∃𝑧(𝑧 = 𝑥𝑧𝑦) ↔ 𝑥𝑦)
43bicomi 209 1 (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 191  wa 378  wex 1692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-10 1965  ax-12 1983
This theorem depends on definitions:  df-bi 192  df-an 380  df-ex 1693  df-nf 1697
This theorem is referenced by: (None)
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